CIHM 
Microfiche 
Series 
(Monographs) 


ICMH 

Collection  de 
microfiches 
(monographies) 


Canadian  Institute  for  t  r.  -al  Microroproductions  /  Institut  canadicn  do  niicroror-roductions  historiquos 


I 


.al  and  Bibliographic  Notes  /  Notes  techniques  et  bibliographiques 


T.:3  ''^s.  'ule  has  attompted  to  obtain  the  best  original 
copy  available  for  filming.  Features  of  this  copy  which 
may  be  bibliographically  unique,  which  may  alter  any  of 
the  images  in  the  reproduction,  or  which  may 
significantly  change  the  usual  method  of  filniing  are 
checked  below. 


D 

D 

D 
D 

n 
n 

D 
D 

n 
□ 


□ 


Coloured  covers  / 
Couverture  de  couleur 

Covers  damaged  / 
Couverture  endommagee 

Covers  restored  and/or  laminated  / 
Couverture  restauree  et/ou  pelliculee 

Cover  title  missing  /  Le  tit'-e  de  couverture  manque 

Coloured  maps  /  Cartes  geographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)  / 
Encre  de  couleur  (i.e.  autre  quo  bleue  ou  noire) 

Coloured  plates  and/or  illustrations  / 
Planches  et/ou  illustrations  en  coulei:r 

Bound  with  other  material  / 
Relie  avec  d'autres  documents 

Only  edition  available  / 

Seule  edition  disponible 

Tight  binding  may  cause  shadows  or  distortion  along 
interior  margin  /  La  reliure  serree  peut  causer  de 
I'ombre  nu  de  la  distorsion  le  long  de  la  marge 
interieure. 

Blank  leaves  added  during  restorations  may  appear 
within  the  text.  Whenever  possible,  these  have  been 
omitted  from  filming  /  II  se  peut  que  certames  pages 
blanches  ajoutees  lors  d'une  restauration 
apparaissent  dans  le  texte,  mais,  lorsque  cela  etait 
possible,  ces  pages  n'on'  pas  ete  film6es. 

Additional  comments  /        v.inmr.  |>.ii|iiu)s. 
Commentaires  suppl6mentaircs: 


L'lnstitut  a  microfilme  le  meilleur  exemplaire  qu'il  lui  a 
ete  possible  de  se  procurer.  Les  details  de  cet  exem- 
plaire qui  sont  peut-etre  uniques  du  point  de  vue  bibli- 
ographique,  qui  peuvent  modifier  une  image  reproduite, 
ou  qui  peuvent  exiger  une  modification  dans  la  metho- 
de  normale  de  filmage  sont  indiques  ci-dessous. 

Coloured  pages  /  Pages  de  couleur 

I I    Pages  damaged  /  Pages  endommagees 


n 


Pages  restored  and/or  laminated  / 
Pages  restaurees  eVou  pelliculees 


Q    Pages  discoloured,  stained  or  foxed  / 
Pages  decolorees,  tachetees  ou  piquees 

Pages  detached  /  Pages  detachees 

1/      Showlhrough  /  Transparence 

□    Quality  of  print  vanes  / 
Qualite  inegale  de  I'lmprossion 


□ 


Includes  supplementary  material  / 
Comprend  du  materiel  supplementaire 

Pages  wholly  or  partially  obscured  by  errata  slips, 
tissues,  etc.,  have  been  refilmed  to  ensure  the  best 
possible  image  /  Les  pages  totalement  ou 
parliellement  obscurcies  par  un  feu:!let  d'errata,  une 
pelure,  etc.,  ont  ete  filmees  a  nouveau  de  faijon  a 
obtenir  la  meilleure  image  possible. 

Opposing  pages  with  varying  colouration  or 
discolourations  are  filmed  twice  to  ensure  the  best 
possible  image  /  Les  pages  s'opposa.t  ayant  des 
colorations  variables  ou  des  decolorations  sont 
tilmees  deux  fois  afin  d'obtenir  la  meilleure  image 
possible. 


ThitHMiltatHmtdMtlWrMhiCtion  mtio  checked  below  / 

C«  dOeunMnI  Mt  filmA  «U  laui  de  reduction  mdique  ci-aeitogi 


lOx 

14x 

18x 

22x 

26x 

30x 

1 

1 

/ 

12x 


16x 


20x 


24x 


28x 


32x 


The  COPY  ^'In-ied  here  has  been  reproduced  thanks 
to  the  generosity  of: 

National    Library   of   Canada 


L'exemplaire  film6  fut  reproduit  grace  ^  la 
g^n^rositd  de: 

Bibliothcquc  nationalc  du  Canada 


The  images  appearing  herp  are  the  best  quality 
possible  considering  the  condition  and  legibility 
of  the  original  copy  and  in  keeping  with  the 
filming  contract  specifications. 


Les  images  suivantes  ont  6t6  reproduites  avec  le 
plus  grand  soin,  compte  tenu  de  la  condition  et 
de  If  ner'etd  de  lexemplaire  filmi,  et  en 
CO.     jrmit*  avec  les  conditions  du  contrat  de 
filmage. 


Original  copies  in  printed  paper  covers  are  filmed 
beginning  with  iha  front  cover  and  ending  on 
the  last  page  vAit.     i  printed  or  illustrated  impres- 
sion, or  the  ":       .  c    .er  when  appropriate.  All 
other  original  copies  are  filmed  beginning  on  the 
first  page  with  a  printed  or  illustrated  impres- 
sion, and  ending  on  the  last  page  with  a  printed 
or  illustrated  impression. 


Les  exemplaires  originaux  dont  la  couverture  en 
papier  est  imprimie  sont  filmos  en  com.nencant 
par  le  premier  plat  et  en  terminant  soit  par  la 
dernidre  page  qui  comporte  une  empreinte 
d'impression  ou  dillustration,  ioit  par  le  second 
pint,  selon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  filmAs  en  commenijant  par  la 
premiAre  page  qui  comporte  une  empreinte 
a'impression  ou  d'illustration  et  bn  terminant  par 
la  derni^re  page  qui  comporte  une  telle 
empreinte. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol       m    (meaning  "CON- 
TINUED"), or  the  symbol  V  (meaning  "END"). 
whichever  applies. 


Un  des  symboles  suivants  apparaitra  sur  la 
derniire  image  de  cheque  microfiche,  selon  le 
cas:  le  symbole  — ■♦-  signifie  "A  SUIVRE",  le 
symbole  V  signifie  "FIN". 


Maps,  plates,  charts,  etc  ,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  ;n  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
"lethod. 


Les  cartes,  planches,  tableaux,  etc.,  peuvent  etre 
film^s  i  des  taux  do  reduction  diff^renis 
Lorsque  le  document  est  trop  grand  pour  etre 
reproduit  en  un  seul  cliche,  il  est  film^  ^  partir 
de  langle  supirieur  gauche,  de  gauche  ^  droite. 
et  de  haut  en  bas,  en  prenant  le  nombre 
d  images  nicessaire.  Les  diagrammes  suivants 
illustrent  la  m^thode. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

MICROCOPY    RESOLUTION    TEST    CHART 

ANSI  ond  ISO  TEST  CHART  N.      . 


1.0 


I.I 


1.25 


■      ||||J£ 

'•4     il.6 


A     APPLIED  IIVMGF     inc 


A  TRHAFISE 


y 


ON 


HYDRAULICS. 


HI-  \KN'    T.     I'OV  KY. 

M  'n-i.  r.r. .  i.L  r) .  v  k  s c, 

Pmjeitor  cf  Ciril  Kntinetrin^  and  Af^litd  Mrchan: 
McGill  Vnivtrsily,  Montttal. 


Si- COX n    i-iiir:o.\.    ki-  urit ii-m. 

HKSr     IHIir-AM>. 


\F\V    YORK  : 

Y^\\\    WII.I'.V    .S;    SONS. 

I,(imm)n:    CllAI'M  an    \     HALL,    Limiikd. 

1  (jO  I . 


Copyriglu,   iSj;,   k)oi, 
ilKNKV     i      HDVKS. 


ROW*'    rH%imi^.,0,    »»1NTI»     hi*    /OHK. 


PREFACE. 


'I'm  ])^•-^l•nt  trcati-c  i-,  tlic  niitiiMiu-  .if  lectures  delivered 
in  Mc(iill  University  (luring;  tlie  last  ten  or  tweKe  year<.  and 
altiioii;4h  intended  primaril\- lor  the  u^e  and  eoin  enienee  of  the 
--tudent  of  ii_\clraiih\ -.  it  i-  hoped  that  it  nia\-  also  prove 
accepcahle  to  the  eiiLjineer  in  general  practice. 

hi  order  to  render  the  treatnient  of  the  subject  more  coin- 
!>lete,  tree  retereui  e  has  |,ecti  made  to  standard  authors  i.u  the 
-iihject.  Tile  examples  iiitr.,du>  ed  Im  illustrate  the  text  have 
also  been  ^elected  in  part  from  the  works  ,,f  such  w.I^known 
writers  as  Weishach.  <  )diort!e  Reynolds,  and  Cotterill.  hut 
the  j^rreater  luunher  are  -^uch  a-  liaxe  occurred  in  the  course  of 
the  author'-,  own  experience.  The  t.diles  c.f  coefficients  of 
discIiar^M-  l-.,i\-e  l)ein  pn.pai-i.  d  t'i'>ni  die  lesult--  <<(  expeiiments 
carried  out  in  the  Hydraulic  I  ..ilx  .latory  of  tne  rniversits-. 
I  he  .e  experiments  are  still  hein;,;  lontmued  and  ina\  prohahU" 
form  the  subject  '>f  a  special  p.i])er. 

1  he  author  desires  to  at  know  led-e  manv  -u^^L^estions 
ollered  b_\-  .Mr.  namfi.r<l,  and  to  expre--,  hi-  deej)  ol.li-,ni,,n 
to  I'rofessor  Chamiler  for  much  l.ibor  and  time  ,;i\-en  to  the 
revision  of  proof  sheets. 

lllMO     '1  .    H.  iVI  V. 

M' 1%  I  Kl  A  r..    NoVPIllliri  ,    lSi^5, 

iii 


PRi:fAc:K   To   SHCONI)   HDITKJN. 


'1  III     pri-ciit    edition   <.f  tlir    work    ..n    ■  ■  I  Kdraulic-^  '      1 
been    ]ir.utK,ili_\    icvM-ittcii,    tlu:   xarioii-   MKiiitcrs    li.uui''    1 


la-- 


i,i\  uij^    i)(tn 


rtarr, 


uiLjrii  and    HI  -onie   lasus  (.dm])Icttl>   alt.rt'd    in    <iid 


allow  of  ntH  (.'ssar\'  coriTctidn--  and  of  tin    niti-od 


cr    ti» 
nrtic.n  <  ■!   nmi  h 


nc\\   matter. 


in  (  iKiptiT  1.  arti'.  le-  on  tlic  whniin-  and  rotatmn  ,,(  ilm,!- 


ha\c    ',)(.en    inserted,  tlu-  artii.  U 


W 


rir-^  am 


1   Xottlir-;  '     h 


])ccn  compli-tL-lx-  rewritten,  and  there  has  been  added  a  r( 


^(U 


n/iii  ■-  e\]ierimental  wnrk    on  w cir.- 


uhieh  ai)]iear-  in  tlu'  .]/ni,r/,s  ,/,s  I'oiits  </  C li 


a  emnplete  aeenunt  of 


(!//SSl'l  s. 


In  (,h,i]iter  1 1  w  ill  he  fonnd  a  I 


ir-e  anicamt  mI'  new  material 


niehidm''   the  re-^nlt--  of  e\| 


jerinunt'- eollahoiated  and  talnilatod 


!)>■  Mr.   V  .   W.    I  ntton    of  lUilTalo,  to  whom    1 


thank'^    fur    \arious    n^^■hll 


aHo   n\se    man\- 


rejirc-entation  nfthe  results  of  the  pii 


-u^-e-^tiMn-,    and    ti.r     the    L:ra]ilneal 


)e-liow   exjierinu-nt- 


l  ha|)ter  III  ha-  been  eonsiderablv  ehan-ed  and  lem^thcnt-d. 
'!  he  reMilts  of  the  experiment^  l._\  Ha/in.  (ian-nillet  and 
Kuttir.  and    ..ther-^    are    -i\en    in  detail    and  tabl 


lilies  ,,|'  tl 


le   constants   ui  the  se\iral    sta!idard  f 


es   L;ninL;    the 
orninhe,   Ixith 


in    luiL,'lish    and    nutrieal 


1     units,   are    added    at    the    end    .  it    th 


hajiter. 

Chapter     !  \' 


.  ontains     new     artieles      nn 


a(.  euniulat.  ^r- 


]irtsscs,  and  w,itcr-cnt;in 

Chapter  V  has  been  toniiilet 


|>letely  rewritten  and  miw  includ 


rRF.h.'icH.   TO  si:co\i>  ininox. 


a  discussion  of  tlic   an.ilysis  of  the   imp.ia,  Horda.  cciitritu:.,'al, 
ami  otlicr  turhiiu  -^ 

Ciiaptcrs  \'\.  \'il.  and  \' 1 1 1  in  the  new  \  111111111'  replace 
rh.iptcr  \'ll  of  tin-  old  vnlumc.  I  h.iptri-  \'l  dcai^  c\cl;isi\cly 
uitli  ualcr-uhccls;  Ciiapter  \'ll  ciMitain-.  new  inatti  r  and 
treats  of  the  \  arious  c!a-^SL>  <>(  lui  bine--  \\  hit  h  lia\e  n^t  l.eeii 
<lcalt  with  in  Chapter  \'.  Cha])ter\"Ill  i-  cntiiel_\-  nev.'  and 
deals  with  cciitrifu^fal  ])unip>.  Much  oi  the  infnrniatinii  ini.<i;-- 
])orated  in  this  cha]iter  lia^  heen  obt.iiiu.-d  throu^di  the  kuidne-s 
of  iMr.  A.  !•".  II  dl  of  Huston,  uliolias  ^M\cn  valuable  lunts  and 
sut;|.;L-stions  and  ulm  has  also  furnished  iniport.int  practical 
examples. 

It  is  hoped  that  the  lar;4e-  amount  of  new  material  ;md  the 
various  tables  whiJi  ha\e  been  added  to  this  \ohniie  will 
indicate  the  pro.L;re-.s  uhKh  is  beini;  iiiad.e  in  reduun;,;  the 
subject  of  Ihcirauhcs  to  an  exact  suence  and  that  the-e  addi- 
tions, more  espeeiall>-  the  tables,  will  aiid  consiilerably  to  the 
uscfulne^-sof  the  book  for  the  purposes  of  the  jiractical  engineer. 

I  lia\e  now  only  to  express  my  ,L;ratitiide  to  m\-  coUea^nie, 
Dr.  Coker,  for  su<;<jestion--  made  M..m  time  to  time  and  lor  his 
};re,it  kin^lne^^  m  re\ising  the  proof  slieet - 

lIl.NKN     1  .   ])iivi;v. 
Octolicr,  i)Oi. 


CONTHNTS. 


ciiAr'"::R  i. 


CKNERAl     PRINCII'I  K"-,     lli.«      lHKOri;ll    OKIUCES,    OVFR    WURS,    tTC. 


Fluid-moiion  .... 

Ste..dy  motion 

Permanent  regime 

Streamline  motion...- 
M'lion  in  plane  layers. 

L.iniinar   motion 

I» 


ensity 

Compressibility . 
Hea.i    


Continuity. 

Bernouilli's  theorem 

Applications  of  Hernouilli's  theorem 

Rotation  of  a  fluid 

Whirling  fluids  

Ori'tce  in  a  th  .u  ['laic 

Tor  rice  Ill's  the  "iini 

Flow  ihi-.u^li  'irifues  in   vessels  in  motion. 

Fiow  in  frictionless  riipe 

Hydraulic  coefficients 

Tallies  of  coefficients  of  discharm-    . 

.M iner's  inch 

i nversion  of  jet 

'I  inie  of  empi  vi'it,'  or  tiiliIl^:  a  Km  W 

(jeneral   equations  ....    

Loss  of  energy  in  shoik 

Mouthpieces 

KnerRV  and  momentum  i^f  a  jet     

Radiating  current 


i'A<.it 
I 
t 
I 
2 
2 

3 
5 
7 
7 
.   lo^ 

12 

I  7 
11 

22 

24 

2'i 
27 

'-•'> 
3') 
44 
4*5 
50 
53 
?5 
5'^ 
6., 
70 


Vlll 


C').V77:.V7.s. 


FA'." 

\r)rtex  mothiii                         74 

Lir^e  ontices  in  vertical  plane  surfaces 7* 

Ncitchesand  weirs ^i 

Reservoir  sluice  i ''7 

Ha;in's  How  over  weirs ')'> 

L.\jinplcs '"* 


ril.M'ir.K     II. 


H  I  II.    l-Kli    I  IiiN      AMI     lllK-H.OW. 


1  J  1 

1  .") 

i:i 
li) 
13" 
'3' 
132 

131 


Fluid  friction •  •  •  • 

Laws  of  tluid- friction 

Surface-friction  of  [)i|ips 

Darcy's  results 

Kcynoi.s    results 

Critical  velocity 

CoiseuiUe's  results 

Kesistancc  of  ships 

Pipr-ll     ^    assumptions 

Steady  How  in  |)ipc  of  uniform  section 

Influence  of  pipe's  inclination  on  tlow   '3*< 

Formul*  of  Uarcy.  Hagen,   riiiiipp,  KcynoMs.  ru      

Diagrams  of  pipe-flow  

Values  of  . ,  .»,  and  V  in  the  formula  :•  ■;:  r<«'(' 

Transmission  ■>(  energy 

Pressure  due  to  shock 

Flow  in  uniform  pipe  conneciinv;  two  reservoirs 

Lo>se»  of  head  due  10  abrupt  ■  han«;es  of  section,  elbowt,  vaIvr^.  etc. 

Nozzles 

Ellis   experiment  on  noz/les 

Freeman  s  nozzle  experiments. ...  

Motor  driven  by  water  from  a  pipe 

Siphons 

Inverted   siphons • 

Air  in  a  pipe ••• 

Flow  in  a  pipe  of  varying  diameter • 

Equivalent  uniform  main 

Hranth  main"<  of  uniform  diameter. 

Flow  in  pipes  leailinn  from  three  reservoirs  to  a  common  junction... 

Mams  with  any  reipiireil  nunilirr  o(  branchei SOl 

Variation  o(  velocity  in  a  transverse  section WJ 

t.auging  of  pipe-flow • »of 

t  vain  pies • "■' 


14" 
iflt 
15" 
fbi) 
163 
I ''4 
'-4 
177 
1:8 
•  7') 
181 
1*2 
iSt 
1S4 
i8t) 
lis 
iqi 


COS'ThX  fS. 


IX 


I 


CHAIMKK     111. 
Il/nv    (U     WAll.K    -N     "1  t.N    (  HANr:El,b. 


Channel  fl.iw  assi.nipuons 

Steady  Mow  in  channels  of  constant  sri  tion 

Retarding  etTect  ot   air.  etr 

lable  of  slopes  and  nie.in  velocities  of  How 

Form  of  channel  ■  ross-section 

l-iest  dimensioni  f)r  trapez<iidai  channel 

Aqueducts 

Formuhc  of  Prony.  Eytelwein.  Beardniote,  and  Tadini. 

Hazin's  formulae 

Table  of  values  of  .1  and  // 

1  able  of  values    It   >- 

(.".anguillet  >V  K utter  s   fornnila 

Table  of  value-,  of  n 


i  (O 

231 
2H 
237 
223 
233 
340 

a47 
34<> 
349 
350 
350 
3JI 


Formul*  of  Manning,  Tutton.  Humphreys  &  Abbott,  C.ankler 25a 

Variation  of  velocity  in  channel  cross-section 

Hoileau's  formul.T 

Tables  of  erosion  and  viscosity 

Kiver-bends-       .  

(  hannels  of  varying  cross-section 

lables  of  values  of /for  standing  wave 

Longitudinal  proiile  and  Kilhlmann's  law 

Channel  of  rectangular  section  of  small  slope 

Channel  of  great  width  as  compared  with  1  he  dei.th 

I. .hies  of  values  of  l)ackwater  function.  <,'•(:» 

(h.mge  of  section. ..  .  

<;auging  of  streams  and  watercourses 

Determinations  of  mean  velocity  of  How 

lables   of    values    of  coefficients    in   formulif   of   Bazin,  Ganguillet  ti 

K utter,  and  Manning .m-3»7 

hxaniplcs 


as? 

...  368 

. . . .  36ft 

26() 

. ...  371 

....  3RI 

....  285 

....  387 

2B« 

.390.  2g2 

aqi 

297 

21)8 


128 


CIIAI'ITK    IV. 

RAMS.   l'KK-.SK.,    Ac:i:llMI|.AIoK-..   \^  A  I  KR- r»KsSi:i«K    KNOINKS. 

Ilvdraulit  rams '-'•• 

Packing "'' 

llvdraulic   press    3'' 

Uvdraulic    |a.  k i'^f 

I'unching  bear -^3'' 

Accumulators J*'* 


cox  TF  NTS. 


Uiflerential  accumulators ,,, 

t  .  *         •  ■ j-*^ 

.Steam  accuniulatur 

,  *  344 

liyuraulic  engines  .  . 

Losses  of  energy  in  hydraulic  engines ,;, 

"^^'" : '.......'.'.'.'.'.   353 

Example! 


ClIAI'l  IK     V. 
tMT'ACT,   RKACIIDV,    IMIAC  r  AND  H  M;|.  s  n  A  1     IIKHINKS. 

Impact  upon  .i  flat  vane 

a  series  of  flat  vanes jf, 

"  "      surface  of  revoluti<jn 

a  series  of  surfaces 

"  "       a  bordered  vane  

I  111  pact  apparatus * 

Coefficient  of  impait 

Keartlon 

Jet  propeller 

Jet  real  tioii  wheel  (Scotch  turbine) 

In) part  wheel 

Borda  turbine 

KfTert  of  friction  on  impact  turbine  

Danaldcs 

Tub  wheel 

Impact  on  a  (  urveil  v.ine 

Tangential  ((enlrifugall  turbines 

Jet  turbine , 

Resistance  to  motion  of  a  solid  in  a  fluid 

Pressure  on  a  thin  plate , 

I'ressurc  on  a  cylindrical   body  

Examples 


a 

3f'^ 

36.J 

37-' 
371 
37J 
37; 
37-f 
3»i 
384 
3*', 

3S7 

i')} 

40  1 
40J 

4'>4 
4.rf, 
4u4 


Clt.APTF.R    VI. 


VrK  I  II  AI     \VA  IKK-WIIFI't  s. 


Classilii  ation    . 

Undershot   wheels 

Wheels  in  straight  rare 
Losses  in  straiKht-ru(  e 
Mechanical  cflei  i  of  itraiKhi-race. 
Poncelei  wheel 


....  4K. 
.  . .  .  4i'> 
....  41-1 
...  42' 
•A2i>,  4»< 
....    u* 


CON  r  f:\Ts. 


Mechanical  effort  of  I'onrclct  wheel . 

Elikiency  of  I'uncelct  wheel 

Ki)riti  of  buckets   

Sluices 

Hreast-wheels 

Speeil  of  wheels 

Mechanical  effec  I  of  wheels 

Siigebien  wheel 

Overshot  wheels 

Velocity  of  wheels 

Kffect  of  centrifugal  force  in  overshot  wheel. 

Weight  of  water  on  wheel 

Arc  of  ilischargi- 

Capacity  of  bucket 

Us^eful  effect  of  overshot  wheel 


.42■^ 

435 


•452. 


428 
4J2 
458 
4.17 
44' > 
411 
44^; 
44'^ 
4V' 
450 
451 
452 
4=-7 
458 
4''7 


(  (I.AI'I  r.K  VII. 


Reaction  ami  an puise   turbines 5 

Actual  path  of  a  tiuitl  panicle  in  a  turbine 

("lassiTaation  of  turbines 

Analysis  of  turbine 

Practical  cocrticients.... 

'rhrory  of  draft  tubes    

Losses  and  nicchaniciil  effect 

Exan^ples  . 


48(. 

4V<) 

4')7 
5l'> 
52'» 
531 
57<> 


I   II.M'll  K     \  111. 


"KN  1  KIHi;AL    I'I  MI'S. 


54T 


(icnrral  statenirni 

Analysis  of  cenlrifuKal  pump 

Losses  in  hydraulic  lesisiance   "  ,,, 

HIade  angles  , 

Volute        ....',"......'."...." 

Whirlpool  I  hanilici 

■  ''■artM  al  ciietficienti 

Kxaniiilrs  


554 

5f<' 
558 
5ftS 
5''<l 
57« 


HYDROSTATIC    PRINtTl'LT:S. 


FVMiAMI  N  1  Al  I'RlN'll'M-  ■'!  H  VKKOs  lA  I  l( -.  — //«/,A  may  be 
divi(if<i  into  two  <  lassfs  : 

I  iijuiil<.  wliicli  are  iiic.nni-i'ossiljlc,  or  iicarlv  so.  sliowirij.;  i:o  sensible 
<'liaii^e  of  volume  muler  (  liaiii;i's  of  pr<N'.urc.  and 

(iixses.  which  ari'  coiiipros^ihlc.  c  hani^iny  in  \iilmnc  with  (  haiit;cs  of 
[irtsMire. 

Tlic  pressure  of  a  perfect  (lunl  on  anv  surface  with  w1p(  li  it  is  jucoti- 
tai  t  IS  perpendicular  to  the  suifaie 

The  pressure  of  a  fluid  at  anv  |)oiiit  of  a  surface  is  tlie  pressure  per 
tmil  of  area. 

The  pressure  .it  anv  point  of  a  tliiid  i^  the  same  in  every  direction. 

Any  pressure  apfihed  to  tlie  surface  of  .!  fluid  i>  liansnutteif  e(|ualiv 
til  .ill  ji.irts  of  the  fluid. 

The  <lensity  of  any  uinf'-rin  snbstanee  is  the  m.is~  if  ,i  iiiiil  of  volume 
of  till'  substance. 

The  intrinsic  weight  of  a  substance  is  the  weight  i.f  .1  unit  of  volume 
•  if  ihi-  substance,  expressed  in  terms  of  some  standard  unit  of  weitjiit. 
The  difference  in  the  unit  due  to  change  of  locality  is  very  slijjht  tin- 
ratio  of  polar  to  eipiatorial  gravity  beinj;  32.2527  :  32.08S. 

The  specific  gr.ivitv  of  .1  siibst.mce  is  the  ratio  of  the  weieju  ,if  anv 
Volume  of  the  substance  to  (lie  wci};lit  of  an  efjual  \-oliime  of  .1  standard 
siihstaiue. 

It  fluid  volumes  \'A'  .V" — of  densities^!.  i>' .  n'  ,ire  iinxeil  to)4eti.er, 
the  density  of  the  mixture  =  '^(oV)  +  i'{  ('). 

If  fluid  volumes  f,  /".  /"'—of  specific  Rravities  .v.  5',  .t" — are  mixed 
tonelhcr.  the  speciti(  uravily  of  the  mi.xtiire  —  •2;"(«t')  -i-  i"(  {•'). 

The  pressure  in  a  homoRi'iieous  fluid  at  rest  under  jjravity  increases 
uniformly  with  the  depth,  nr.  m  other  words,  the  ditlereiice  of  the  pres- 
sures at  any  two  p<jints  varies  as  the  vertical  distances  between  the 
jioints. 

xiii 


MV 


H)i>R( )sr-iTi(:  rKixrjpiFs. 


.\nalytii\rlly,  the  (iill(renc<'  of  pressure  —  :r~.  :f  boiiii;  the  intritisir 
\veii;lu  of  tlie  fluid  and  s  the  ditlereiice  <if  level. 

The  free  surface  of  a  litjiiid  at  rest  under  gravity  is  a  horizontal  plane. 

The  eoniiiion  surface  of  two  liquids  of  dilTerent  densities,  which  do 
not  Tuix.  is  a  horizontal  [ilane,  when  at  rest  under  f;ravity.  If  a  number 
of  li(iuids  of  dilYerent  densities,  e.ij.,  mercury,  water, and  oil,  are  poured 
into  a  vessel,  they  will  come  to  rest  with  their  common  surfaces  horizon- 
t.d  planes,  the  densities  of  the  liquid  increasin<;  downwards. 

The  surfaces  ni  e(jual  pressure  are  liorizont.d  pl.ines. 

The  pre-sure  of  a  liquid  on  any  horizontal  area.  . /.  is  ciinal  to  the 
Weight  of  a  column  of  the  !i(]uid  of  which  the  area  is  the  base  and  of 
whi<h  tlie  heitjht,  c.  is  equal  to  the  depth  of  the  area  below  tlie  surface, 
i.e.,  w.lc  (disrei^.irdinj,'  the  pressure  on  the  free  surface). 

The  whole  pressure  of  a  fluid  on  a  subnierj^ed  suilace  is  the  sum  nf 
all  the  normal  pressures  exerted  by  the  lluid  on  every  portion  of  the  sur- 
l.ice  and  (di>refjardin}»  the  pressure  on  the  free  surface)  is  equal  to  the 
\veii;ht  of  a  column  of  liquid  of  which  the  base  is  equal  to  the  ,irea  of 
the  surface,  and  tlie  heifjht  is  equal  to  the  depth  of  the  centroid  of  the 
surface  l)clow  the  surface  of  the  liquid.     Thus; 

I,/)  The  total   noim.il  pressure  on  a  wall  of  width  /»,  sloping  ,it  "  tu 

tlie  verticil  .uid  lelainirig  w.iter  which  rises  over  a  ientjlh    -  of  the  w  ill 

J  m^c^cos '/ 

=  :c'i-      Cos  0  -_- 


(/'I  The  total  ]iressureon  a  iin'iilar  v.iive  of  dianiet<'r  ./,  with  its  cen. 

JT  i^ 

lid    -  liilow  the  surface  =  ri'     -    z. 

4 

(<  I  'Ihe  total  notm.il  pressure  on  a  loc  k-L;.ite  e>l  width  //and  on  whuh 


the  w.iter    rises  to   a  heif;ht  -  =  whz' 


u-lsK 


The  |iies-.urc  hftivien  <t  pair  of  lock-gates  =  pressure  on  the  hiiii;i' 
|)ONt  —  YiVhL^  sec  (r,  za  being  the  ani^le  between  the  j^.ites. 

The  Centre  of  pressure  of  a  plane  area  is  the  point  of  acti<m  of  the 
resultant  fluid-pressure,  (A'l,  upon  the  plane  area. 

If  r,  ~  are  tiie  horizontal  and  verticil  distances  of  the  C.  of  I'  n  ■■• 
the  vcrtic.i'  and  liori/untnl  axes  through  the  t".  of  (J.  of  ijie  area 

_«'/^_    "'/>   _     /)  ,-_'"/    ^  >jjk'  _   k' 

''   ""     Is      -    -vAl'l    "    Air  "    "     A        '   r.,,1/,    ~    //  ■ 


/>  being  the  product  of  inertia  about  the  axes  ;  /  the  moment  of  inerti.i 
of  the  area  about  Ihe  axi«  off,  //  the  depth  l>el<)W  the  surface  of  the 
centroid,  and  i-  the  r.idius  of  j.;yration. 


//  YDROS  TA  TIC  PK!\aPJ.  i:S. 


XV 


Ex.  I.  Depth  of  C.  ui   P.  iA  ,i  patiillcl.j^i.im  wilii  one  c(\r;c  in  surface 
=  3  of  (ioptli  of  opposite  C(1i;l-. 

i:.\.  :.  Dcplh  ol  C.  ol  P.  oi  .1  tri,iiii;ul,ir  area,  tlie  nii<i(lli-  points  of  the 


side;- b-iti-  atilcptl's  ,/,  .  ,/,, ,  ,/.,  below  the  suifare    = 


'/i'  +  (l-/  +  III- 


"i    +  111   +  iij 
lit'   Ui)  if  veilex  is  III  surfa(e  and  iiasc  hoi  i/mumI,  depth  =  ]  ,,f  depth  (,f 
base  ; 
(i)  if  base  is  m  surface,  depth  =  J  of  dcptli  of  emtex  ; 
(.1  if  vertex  is  m  siirlaee  and;   .mil  .:  .ue  d<-ptli>  of  ends  (if  base,  the 

I  y  —  c 

)•■'  -  c'  ■ 


depth  = 


Tlie  resultant  pressure  on  the  surface  uf  a  <:olid,  wliollv  or  partially 
iininersed  in  a  fluid,  is  equal  lo  the  wei-ht  of  the  displaced  liquid  and 
acts  vertically  upwards  in  a  line  jiassiiij;  through  the  centroid  i.l  the  dis- 
placed liquid.  In  other  words,  a  solid  iniinersed  in  a  li(|iiid  .ip[ie.irs  to 
lose  as  much  of  its  wei-ht  as  is  equ.d  t  i  the  wei-ht  .  ,f  the  fhnd  it  displ.ices. 

If  a  homogeneous  body  float  in  a  luiuid,  its  volume  will  bear  to  the 
volume  immersed  the  inverse  ratio  of  the  specilic  ^,'ravities  of  tlie  solid 
and  liquid. 

A  body  of  wei-ht  If.  <  arrvin-  ,i  hud  /'.  I|  .its  m  ,i  Ikimii!,  ^;  and  // 
bciiit;  the  ceiities  of  i;ravity  of  the  biMh  and  .,f  the  dispLued  w,it.  r.  so 
Ihal  <;// is  verticil.  If  the  |i,.id  /'  is  sl-itcl.  the  body  will  heel  lhr..u-h 
an  aiijjle  0  am!  the  punt  //.  aisD  called  tlie  c-ntre  of  buoyancy,  u  ill 
move  on  a  curve  or  suiface  of  buoyaiu  y  to  ,i  new  position  //'.the  line 
'>■//  coiiiiectint,'  //'  with  the  ru-w  p.isiii.ui  (,f  the  C.  of  (,.  of  the  body 
beitin  vertical.  If  »  is  small,  the  iiltniiate  i.osinoii  of  ,1/.  the  mterse.- 
l  ion  of  //(/'and  //'(;',  is  called  the  «;i'/,7, , ///rr,  and  .1/  is  i  ni'refi  >i  e  I  he 
centre  of  i  iirv.iiiire  of  the  surface  of  biiov.ini  v  .it  //.  I'or  sLibditv  of 
equilibrium  .1/  must  be  above  (/.     Theoretn  ,ilU-, 


//.I/ 


«■/  _  «'///{•' 


"r  ' 


W  beiiiK   thi-  w.it.-r  line  are.i   ..ml    ftli,-   v.linne  ..f   h.piid    displ.iced  by 
body. 

C,M-ll,l,AkV  PUKNOMI  \\.-If  ,1  (.Kiss  tube  of  h  Me  hoie  ,s  pl,,r,.,l  verti- 
c.illy  in  a  liquid  like  water,  winch  wets  tli-  >;l.;ss,  the  water-surf.ice  on 
the  outside  ne.xt  the  glass  is  cievated  .in. I  siiilitU  <  on,  ave.  while  ,,n  the 
inside  the  water-surface  is  concave  an. I  there  is  a  marked  elev.ition  above 
the  outside  surfaee. 

With  a  liquid  uhi.  hdoes  not  wet  tti.'  glass,  l,k,<  m.-rciirv.  an  opp.witr. 
effect  is  observed.      There  is  a  depression  on  the  outside  ,,nd  the  surface 


//  YDR(  ).S  TA  TIC   PKIS  t  in  HS. 


IS  sli),'htly  convex,  wiiile  '>ii  tlie  iiisidc  the  Mirlace  is  convex  ;inii  ther.-  i^ 
a  iiuirked  (iepnssion  Dclow  the  outside  siirfiuf. 

SUKFA(  I-,  TKNMON-.--At  the  ix.uiuhn-  surface  sep.initint,'  .iir  imnx 
any  hq.iid,  (,r  hetwee.i  f.vn  lujuids,  then-  is  ..  suiUk  elensiun  winch  .. 
the  same  at  every  p.. ml  ami  m  r\i-rv   iliie(  tii.ii. 

At  tiie  iine  ol  jii.,ction  of  tlie  Ijoiie.dini;  surface  of  a  k'hs  and  a  \u\ui>i 
witli  a  soiid  txidy.  ,jr  of  tlie  l;ouMdin),r  surlace  of  two  liquids  wuli  a  solid 
tj<jdy,  tlie  surface  is  inclined  to  the  surface  of  the  solid  hudy  at  a  definite 
angle,  depending  upon  the  naliiie  ul  the  solid  and  the  liquids. 

1  lie  surface-tension  is  independent  of  the  -urvature  of  the  surface 
but,  if  the  temperature  be  increased,  i'  diminishes. 


USEFUL   CONSTANTS. 


Thf  following  abbreviations  are  used: 
uhic  metre  :=  in.';  centimetre  =  cm.;   scj. 


metre 


gramme   —  k. 


;     kilometre   ^    kilo.;     gi'^in    ~    tjr 
kilo){ramine  metre  =  km. 


Metre  =  m.;  sq.  metre  —  m.'; 
entii,ietre  =  cm.*,  cubic  centi- 
t;ruinnie  •-=  ijni. ;     kiio 


1    ;i;. 

—  2  ^4  cm. 

I    rm. 

-  -39370"  in. 

1   (I. 

-  3"-  '799  cm. 

1   in. 

•=  3.280843  ft. 

1  mile 

"    i.6of)3  kilo. 

I   kiit  . 

.(12137  mile. 

I   ktii  I 

!   nam.  mile  per  hr. 

=  (KiSo  ft.  (av. )  per  hr 

1  ■-().  in. 

—  6  4^  Id  1  m.' 

I  <  m.* 

=  .155  sq    in. 

1    S(|.    ll. 

-  929.03  cm.* 

1    Til.' 

-    10.76311  SC].   ft. 

I    "-ll,    Vll. 

=    .83612(1  Ml.' 

1  a<  rt 

=  43.?0-i  K(|.  It 

-  .40468  hectare. 

I     III.   I.iTJ 

10,(100  m.' 

loi  .ire*.. 

—  2  471 '  ai  res. 

I  •■<).  mile 

—  640  acres. 

-  2.59  s().  kilo. 

=  25<)  hectares. 

I  "-ij.  kilo. 

—  loo.cxxi  m.' 

=  24.71 1  ai  res. 

1  Hi. 


1  k 


-     16  1)2.    rr    7000  (Tf. 

=  ■453?924  k. 
=  453- 5924  urn. 
=  445,000  dynes. 
—  3  204622  lbs. 
=  981,000  dynes. 


1   Hriiish  till 
I   I'.  .S,  toil 
I  !"r.  tonne 


2240  lbs. 
-   1016  k. 
=^  20<Ml  lbs. 
=--  9f'7i43  k. 
=  lixio  k. 
=    .i;342  Hritlsh  ton. 

2204.622  Ihs 


J  cu.  in.  i.f  u  .i!«T  at  4    C. 
I  cm.'" 

I  I  u.  ft. 
1  litre 

I  im[i.  sal.  at  ti2     1'. 
I  cu.  ft.  of  water  at  62'  K. 
I   I  11.  ft.  i  if  air  at  O"  C.  t 
and  I  atm.  > 

I  cu.  ft.  (if  hydroyeii  at  / 
1.    C.  and  I  atm.  1 

I    litre    I  if   air   at  o    ('.  ^ 
and  1  aim.  ( 


=  252-89  >;r. 

-  I  Km. 

~  62.43  lbs. 

-  I  k. 

-  10  lbs. 
--62.3  lbs. 

-:      .OSO7    lb. 


•  .2')32Km. 

th    of    its 


Water    compresses 
bulk  under  a  chanRe  of  pressure  of 
I   it.n..  or  about    '„lh  of   its  vol.   un 
der  ,1  pressure  I  if  2  tons  (of  2240  llis. ) 
Iier  •..!.   :n. 

I  lb.  per  s(j.  in     —    0703  k.  per  rm.'-' 

I  k.  per  cm.'  .0703  lb.persi|.iii. 

I  lb.  per  s<|.  ft.     —  4.8826  k.  per  ni.' 

~  47()xlynesperrni.' 

xvii 


X\lll 


Lsr.hLL    Uh\S7.-t.\rs 


No.  of  lbs.  per  ) 
sq.   in.  f 


No.  o(  k  [,,.r  m.' 

I  in.  of  mer-  ) 
<  ury  at  o"  C.  f  " 

I  rum. mercury  i 
at  o   C.  )  " 

I  cii.  in. 
I  rni.-' 
I  cu.  ft. 

I  m'. 
I  litre 


=  l4-2::>  k.  per  rni. 

.  (  -A'jo-;     ins.      of 
(      mercury, 
j  in.s.  of  men  ury 
I        ■    2.037.S. 

_  (  4  S-2f    Ihs.    per 
'        SC|.  fi. 

-•034534  k.  pertt!!.- 

-.o<)i-!5()(,  per  cm.- 


I  imp.  fjal. 


I  U.  S.  gal. 


^  at    (irreiiwi(  li 
A'  at    London 
J,'  at    M.mi  tirster 
.?•  al  llic   i-(|iiator 
,C  at    Haltiinore 
.fat   Montre.il 

The     inertia    or  1 
mass  of  a  body  \ 


=  "'  3.-'7  cm.' 
=  .of)t  cu.  in. 
=  .028317  m.» 
=  28.317  litre.':. 
=  35.3:4s   cu.   ft 
=:  icMjo  cm.' 
=  i."5')S  iiinfi. 
=  .22  imp.  gal. 

—  .1605  <  11.  ft. 

—  277.27  cu.  ins. 
=  4  54^'K'3  litres. 
=  231  cu.  ins. 

=  .83254  itnp.  gal. 
_  (  OSi  cm.  per 
•"~  /  .sec.  per  sec. 
_  j  32  2  ft.  per 
'  sec. per  sec. 
-;  32.19078  ft. 

-  <)8i.i7  cm. 
=  32.182  ft. 

-  <)8<)  ()  cm, 
32.U)t,  ft. 

~  9SI.34  cm. 

-  32"S8  ft. 
=/)7S.o4  cm. 

-  32-152  it. 
=  q8o  cm. 

=  32."7'i5  ft. 
=r  980  73  cm. 

(  its  wt.  in   lbs. 
=  I       at     I.uiuion 

I        -  32.2. 


I  2Q-95     ins.     of 
'       nif-rcury. 


I  stanilaril  attn.  I 
of  14-7  Ills. per  ] 
s<j.  in.  J 

—  760  mm. 
I  metric  atm.  of  I 

14.  223  lbs.  per    frrr   .*    ^^-"^      '"^-       "' 

sq.  in.  j 

I  erg 

I  Kin.  cm. 
I  ft.  n.. 


iry 
I  cm . 


I  km. 

No.  of  ft.  11). 

1  II  T.  v. 

I  k.  (Ic^;rep  C. 
I  calorie 

I  Watt 


~  1  dyne  ,- 

=  9S1  ergs. 

=  .1382";  km 

~ '-35^ 2Xio'ergs. 

=  7-233  ft.il).s. 

=  9.81  X   Kj'ergs. 

~  7  :;i:'^  km. 

-777  !!■  1.  r. 

_  \  I3')<»    lbs.   lie- 
*       grce  C, 

—  105S  Joules. 
=  1058  K  10"  ergs. 
=  4200  joules. 
=  4200  X  10^  ergs. 

-  I  k.  raised  i    C. 
=  426,9  km 

"-  3<>8o,.)fi.-lhs. 

=  I  joule  pt-r  sec. 

(  vvr.rk  done  by 

_  j       .■■■  I  urient  of 

I        I    .imp    al    I 

i       v.di. 

550  lbs,  per  sec. 

_  (  74''  X  10'  ergs 

)      per  sec. 
=  74')  u,ilt< 
_  )  i.o[  f,,,,  cs-de- 

'       clicval. 
_    (  .9S^3        horse- 

t       powei. 
=  73f>  walls. 
_  j  545  f(,  lbs.  per 

I      sec. 
=  75  km.  (>er  sec 
I  radian  -  57-296    decrees, 

lo  convert  common  into  hyper- 
bolic  and  hyperbolic  into  .  ommon 
''>Barithms.  multiply  the  former  by 
23025  and  the  latter  by  .43429. 


I  h. 


irsc-pfiwer 


I  force-<le  cheval 


HYDRAULICS. 


('ii.\r'ri;k   i. 
f;iNi:[<.\i,  PRiN-ripi.i:s.    i  i.cw  tiiroccjh  oRincTs  ovfr 

WKIKS.    I.TC. 

1.    Fluid    Motion.      Tlu-  t.rm   ■•  l.ydrauliV..  "  as   it.  .l.-Hv,- 
ti..n      ro,..„,     water-    ,.r,\/c,    ^    tube    or    pip,-,   „„iKatc^,    uas 
l'nn,.,n!v  applied  t<.  ihf  ecu .  vancc  Mfnat.r  in  a  tulK^  nr  pi,x- 
t)«it  it^  uua.lii-    n.,u    cni'.racc.  tlu-    cxpcniucntal  tluorv  uf  the 
initio!!  (if  t1iii(K. 

■"'^-     """i""     '■'     ^'     nni.l     i.     .ai.l     to     !„■    ./,■„,/,.    .  T    /,,  ;;;.;;,-;,/ 

'A  lull  thv  ni,.Uciik-.  Macc-.M-vcl>-  arrix  in-  at  aiiv  ^iv.-n  p,int 
arr  anMuatr.i  u,t],  ,!„■  same  vlontv,  arc  s,i,,,Vctol  U>  tin-  same 
pressure-.  an,i  are  the  -.urn-  in  .leiiMty.  As  .,,.,„  ,,s  tlu-  nn.non 
"f  a  stream  heomu-..  st,.ulv  a  /,/;//,,«, ;,/  ,,;.vw,  j.  said  t,,  U, 
estahlisiu-,1.  an.l  liv.iranlic  iiu  .■stif^ratiops  .„■<■  „s„al1v  ,,,a,1e  on 
the  hypntheMs  ,,r  a  p.rmanmt  r.-jme.  With  such  an  lnp,,thc- 
s.s.  any  portion  ,,|  the  flui.l  mas..  ^,  !„-,],  |, ,,,.,,,  _,  ^ri,.,.,,  n-j„„ 
I-  replaeed  l,y  a  like  portion  un.ler  eonditions  uhiel,  are  i.lenti- 
taliy    the  '-ami- 

1  he    t<  mis   ••  stca('\-    motion    '   an.l 


arc  otteii  ^onMilereil  to  1.,-  sm,,  .runions. 


permaiu  nt    r<'-;iine  " 


2  liun^  Miinos: 

The  rjcncral  problem  nt  t],<\\  is  tin-  (li-tL-niiiiiatinii  of  tiie 
rclatioti  uiiicli  c\ist>  at  .my  ]ii>mu  hctwii'ii  the  (I(.Ti--it\  ,  prc^- 
•^nrc.  and  \cli-Lil_s  ol  llic  iniilcnilc-.  uliicli  siKCcssivtly  [i.i---'  that 
point. 

1  he  actual  niotinn  of  a  llniil  is  cveci'iJin;;!}"  eomplcx,  and, 
in  ori!i-|-  to  siniplity  the  in\cstiL;ations,  \  arioiw  a-^^iiniplion--  arc 
inailc  a-^  to  tii^'  iiatnic  ■'!  the  tlow  . 

2.  i(/i  Stream-line  Motion.  '1  lir  nioKeuIrs  may  he  le- 
f^ariled  a-^  llow  in^  alon:;  ilehnite  i)atiis.  anil  a  --ik  cession  if 
-^uoh  niolecules  as  forniini;  a  coiitimion->  lliiid  rope,  which  is 
terniid  an  iLiiirutiiry  sti\(V:;  or  a  llmd  filamcni  ;  or.  if  the 
motion  i^  ste.iily,  and  tin-  patli-.  therclorc  fi\t:d  in  sp.ice,  is 
tcriiU'd  a  slriitiii-liii, 

I'-xperiment    shows  th.it    tiic    Nrh.tityof  How    in    ,iny  eross- 


tlie  secti'in  i^  m.ulc  up  ot  ,m  intniite  inimhci'  of  indcfmittiy 
ismall  ,ux'.i>.  e.K  h  .iic.i  Im  iii'_;  the  section  o|  .i  tlnid  id.iment. 

I /'  Motion  in  Plane  Layers.  ^!n  this  motion  it  i-^  assnnu'd 
th.'it  the  mohviili'^,  winch  .it  .in}'  L;i\cn  monu  nt  .ue  fmnd  in  .i 
phme  hi_\ci.  will  reni.un  in  .i  jil.mr  layer  .iltei' they  li.ice  mo\-e(l 
inti  1  an\'  new    p.  i-^i;  i  iii 

(.1  Laminar  Motion,  'hi  tin-  hypothec'-  the  stream  is 
su])po-,t(l  to  coiisi-t  ot'  .m  infun'te  number  of  indefim'tei\  thin 
l.iycrs  I'he  \  ni.ition  in  \elocity  from  ponit  to  point  of  ,i 
cross-section  m.iy  tlien  hi-  .illowcd  loi  ,  by  L;i\in;.;  the  se\'cral 
layers  ihiVerent  \i'l.Kities  h.iscd  upon  'he  l.iw  of  Iliiid  resistance 
between  Lonsi_-cnti\-e  l.iyeis. 

3.   Density;  Compressibility;   Head;   Continuity. 

'i  lie  tree/in_;-point  of  puie  water  is      3.:     !■'.  .11        o"  C 

boiling-      ••      ••  J IJ     !•'.  or  loo"  C. 

in.iN     deiisitv '•       •'      39^  I    I',  or       4    C'. 

st.uid.ird  nuMii  temperatuic    ••     U2     V .  or  \()  .66  (". 

The  comp.ii' iti\  e  densities  and  .tls.i  the  i  omp.ir.itive  \(ii- 
umcs  are  the  s.ime  .it    ', j     i-'.   .:nd  4'>'   I". 


Finn  DFS'SITY.  3 

'I'lic  l)ulk  'if  iVcsh  siiow   is   I  J  tinu-s  tlic  !'ulk  nf  tliu  fiiiii\,i- 
Iciit  water. 

I   HI.  tt.  lit  frc-^Ii  siiiiw  \i,ci;-,'li-;  5. J   \\^^.   .uui  its  s.  •^.  is  .oS^:;. 
I    cu.   tt.  (if  iri.-   at    j:;  '   F.   ucii^lis    57I    lbs.  ami    its    >.   l;.  is 


I    til.  tt.  i>l' ■i\-craL(c  --ca-watcr  ,it  ^o'  !•".   \vcij4l1s  114  jhs.  .md 
it-  a\-(. fa''c  s    '<.   i-.  i.o2iS. 


1    I  II.   It     1  it  iniif  \\  atcr 


I    i  11,   yl. 

I   111.  inctic    •■ 


t    :;  J     1-".   \vci;;hs  62.4  iS  lbs. 

•  yy.X    V .      ■■  <)J.4-\^  ■• 
'  52   -3   1'.      '■  f  I  J,  41)1)  •• 

•  >>2     !■.         ••  <i2.  :;55  •• 
'  2 1 J     !-.      "  S'J-^'40  " 

\    ^''•2355     _i,r;,|lotlS    (II- 

/  ').-WN  imperial  L;aii(>ns. 
1G8.36  j^alldiis. 
JJO.09 


aitaiiis 


'l'!ic\'i|     of  I    lb.    (if  [Hire  u, iter  at    ~2     V . 

V/'.  I    !•" 

•■         ■•  5J  .3  !■• 

•■       ■•  r>j'  i'. 

••            •■  2  12'    I-'. 

'I'lie  \(il.  (if  I  t'ui         ••         •■  5-  •  ^  !■'• 

>r,i-uatoi'  ,ir  ()2     I' . 

I  t'lniu- (>t  |iiii  (•  w  att'i  .It  3(y  .1    !•". 
••       ••          I  kild-. 


S  .01602  [  HI.  ft. 
.  O  I  f)0  1  (J 
.Old 
.016037 
.01677 

35-9 
35 

.35- 3 156 
•0353 


I   L;all-iii  (it  i)un-  uatoi   ,u  'ij     j-'.  \\ri;_;lis   10  ll)s.  and  its  \(>l. 
=  277.123  Lii,   iiiN.  ;=  .16037  cu.  tt. 

I  iiiii)cii,il  ^;,iil(in  (if  piiic  water  .it  62"  F    wci^'hs   10.00545 
lbs.  ami  its  vol  -77-74  ^'"    i"^' 


hiUin   DhSSlTY. 


U) 


IT. 

v. 
u 


i    -u: 


-r  «  ^ 

0   - 

N 

u^  ^'  in 

1^2  u\ 

«  r-  - 

-to 

t^ 

r^%i^ 

.    '-^ 

in  « 

c  m  r 

t^ 

»*•- 

.    X 

.(-■ 

N 

66 

ggf: 

^^ 

m  u^  in 

'*^    3v   '^ 

6  c-  6 

« 

? 

6 

r4 

M   M   ci   .-*   N   «   c*   r*   M   ri   CI   r»   vi  f*%  f ,  f,  e^  c 


1^ 
1  - 

in 
»n 

m 
f^ 

ao 

1- 

'/ 

rr 

^ 

w. 

»^ 

g« 

t'. 

m 

<*■. 

•r 

C 

B  =  £  I  £■  b  -  <<i  -r.  iri  t»ac'  o  P  «  -»  in  e  1  >  ~  O  -  "^  U-.  c  ^  C  :  r<  T  O  C  «  " 
j«|a  •Sr^i>i-r»i-r^  i../^  «  cc  x  it  «  «i  «  5'  c  c-  c-  ?■  C-  *  6  "  O  if.  *  "  3 
1-.=  ^   I  -«Min 


X 

c 

i«a:- 

t/; 

" 

'A 

i'i 

< 

~^^ 

&. 

it 

b. 

i»- 

V 

U 

^  . 

'5  X 

^? 

GC 

< 

tl 

C    0 

c  6 


^  i 

i    S 


;j   N    —      .    ^ 

III  :  ? 


c2  sc 

1? 


10    !*}    ' 

5?! 


i^    3,SSSRR^:;:;5'4'*4"^i?  J.S!.XKS;t;S'iir27S'j3 


I 


i 


ILL  in    (.OMI'RLSSmiLITY.  5 

Tlic  tcinp.Taturcs  in  tlii-  t.ihl.-  nia>-  he  taken  as  absciss;i.\  and 
llu-  Liirrchpoiidin-  v.ilucs  ni  the  tlinc  rcniamiiiL;  columns  as 
ordmates.  Curves  of  conipai  .itivr  .ImMtw  wci-lit  \>v\  cubic 
toot,  and  wci-ht  jht  '^'allon  aix-  tlui-  ohtanifd,  and  tlic  values 
c.irrcspondini;  t(.  any  spccificil  tnnpcraturc  can  he  easil\-  and 
\ciy  accuratcl\-  dctorniinjd  troni  i1il>c  curvc>  In-  direct  nieas- 
urenient. 

The  \vei.i,'lits  jkt  cuhic  mot  in  th-  tahk'  liave  been  calcu- 
lated by  nieaiiv  of  Kanknic-  .ippmxnnate  ti.rniula, 

•''  \oooJ 

63.435  ~    7'^"+  350,000  • 
•i'    bciiij;    tlie    wei-iit     per    cubic    foot    corres].,,ndin^'    t'.    tli.- 
</(^W/^/c  tc-inpcrature   /;  i.e.,  ^6  r  _(- ordinary  tenip.rature. 

Tlie  specific  \vei},'hts  obtained  b>-  tlii-^  rule  for  liie  lower 
temperatures  are  very  e.vact,  but  for  the  hi^rln.r  temi)eratures 
tiles  become  to,.  J,,,-,-  lluis  tlie  rule  ^rjves  59.76  lbs.  as  the 
\vei,L;ht  of  a  cubic  foot  of  pure  \\.iter  at  211"  V  .  v\iule  actual 
measurement  makes  the  \veij,'ht  59. 04  lbs. 

1  he  comjiarative  densities  betueen  o  C.  and  40'  C.  aie 
the  \-aiues  obtained  l)y  (  h.ippiiis. 

Co»i/>,fssi/>iitti~-V\un\s  are  sensible  co-ni.ressihle  und.  r 
heavy  pressures,  and  the  compression  is  proportional  to  the 
pressure  up  to  about  1000  lbs.  (OS  atmospheres)  per  sq  in^  h. 
Grassi's  e.xpenments  indicate  tli.tt  tl..  >  oii)pressibilit\-  of  uat.r 
<limimshes   a>  the   tenperature   increases       \V,uer    o.mpresses 

al>"ut    47i  .nni.onths(i.e.,    -.^^^.^^^^^^    nearly)    of  it. 

bulk  for  each  atmosphere  1  his  is  equivalent,  .ippr..xi- 
niately.  to  a  reduction  of  ,'„  in  the  bulk  under  a  pressure  of 
2  tons  per  sq.  inch. 

If  a  volume  /'  of  a  lluid  is  compresse.l  by  an  amount  J/' 
under  an  increase  J/  of  the  pressure,  then  the  amouut  of  com- 
pression per  unit  of  vol.  i 

-yr  and  IS  called  tlie  cubical  compression        Ihe  r.itio  of  the 


hLUll)    COMPRFSSIBlLir Y. 


TABI  E   UK    KlA--MinV    OK    Vol.VMK   IIF    LiyuiDS. 
(Reduced  Irum  Gras&i's  rebu.is.) 


Liquid. 

Elasticity  oi  Voiume. 

.'       7i7,oi'0,oix) 

Tempera. urc. 

Merrurv  . . 

o" 

C 

W.itrr.    .. 

4'J  0<>),LKK) 

45  ,()!>  1,0)1) 

18' 

Sea- water. 

5-,y(-X3,0(KJ 

Ether 

1 6,;;  30,000 

1^,000,000 

it 
"4° 

c. 
c. 

Alcohol. . . 

i  25.47o,oo<j 
«  23,38o,(X)0 

7   3 
ij.l 

c. 
c. 

Oil 

44,o<p.i.XK) 

-     or    V  —^ 
I  ^  Jl" 


N.  H. — The  value  fo    mercury  iS  probab  v  erroneous. 

iiunnicnt  nt  pressure  to  t    ;  cubic.il  c.xp.iii^inii,  vi^., 

1-^   tciincd  tlic   elasticity  i>r\()lumc.      This  is  .-lensibly  constant 

\\itlim    wiile   limits,    ami   is   ^'cnerally  ili'iiotcii    h\-    tlie   letter.s 

/>..!•   A'. 

TIr-  vertical  distance  between  tlie  free  surface  of  a  mass  of 
'.\atcr  .111(1  an\- (latum  jilaiie  is  c.illed  the /// v.-f/ u ith  respect  ti> 
that  ])lani  It  the  water  extends  di)\\n  in  th(  ie\(l  ef  tin- 
pi. me,  a  pressure  />  is  produced  at  that  level,  .nai  the  wihu  of 
/",  so  lonj,'  as  the  water  is  ,-it  rest,  is  :.^i\cn  h\   tlu   niuati  ai 


w  It' 

7i'  hein{;  the  specific  wcij^ht  of  the  water  and  /„  the  pressure  at 
the  free  surface.  Thus  the  pressure  ma\-  he  measured  in  terms 
ol  the  head,  and  hence  the  expression  "  he. id  due  t<»  pressure  " 
or  "  pressure  he. id.  " 

A  column  of  water  at  <'»-''  I",  and  2. 3093  ft.  in  liei^'ht 
exerts  a  pressure  of  I  lb.  per  scj.  inch. 

A  column  of  w.iter  at  62"  I',  and  3,V047  ft.  'ir  10.347 
metres  in  hei^jht  e.xerts  a  pressure  of  14.7  lbs.  per  sq.  in.,  or 
one  atmosphere. 

A  column  of  water  at  02  V.  and  1  ft.  in  hci^'ht  exerts  a 
pressure  ni".433  lbs.  |)cr  sq,  inch. 


HF.Ab   or   WATHR.  7 

//,■,„/. A  luad  of  w.iter  is  a  source  of  oncr.L:)-.      A  xulunie 

of  water  (Ic-onuliivj.  fr'.m  an  upper  to  a  lower  lc\(  1  iiia>-  lie 
e:n])loyeil  to  ilri\e  a  macliinc,  wliieh  receives  enert;)-  troin  tile 
watv  and  a-aii)  iitili/es  it  in  overcomiiiL;  the  re-i-tance->  ot 
other  machines  iloiiv_(  uscliil  work. 

Let  O  cu.  ft.  of  w  itir  i)er  secotxl  tall  thri)ii;,Mi  a  vertical 
distance    of   //   ft.        1  hen    the   total    power   of   the    fall    -;  :cO/e 

ft   -ll)s      ="'"''  h.i).,  .-I'heinL'  the  uei-ht  of  the  water  in  pounds 
550 

]ier  ».  uhic  foot. 

Let  A'  he  the  proportion  of  the  total  power  wliich  is 
ahsorbeil  in  o\erconiiiv^  frictional  ami  other  resistances.  I  hen 
the  iffcitivc  power  of  the  fall  ;  \^'Qhi\  —  A' 1,  and  the  effi- 
ciency is  I  —  A 

Coutii'.uilv.  Ima.^ine  a  Ixuindini.,'  surface  enclosin<^  a  space 
of  nivariahle  volume  in  the  midst  of  a  movin^^  mass  of  fluid. 
The  iirinci])Ii-  of  i  ontinuit\-  alfuni--  th.U.  in  any  interval  ot  time, 
the  tlow  into  the  space  nuist  he  e<iiial  to  the  outllow  durini,'  the 
same  interval.  tMvini,'  the  inllow  a  positive  and  the  outllow  .1 
ne}^:afi\e  --i-ii,  tlie  priMcii)le  may  he  exjjresseil  sv'iiibolicall)-  by 

2:C'    -  o. 


11'' 


be  preservetl  s 
an    held  in  solution 


'I  he  continuity  of  a  mass  of  watir  will 
as  the  pressure  exceeds  the  tension  ot   tin 
li   IS  on  account   of  the   pressure  of  tin-  air    that    pump  -  <  aiiiiot 
draw   water  to  the  hdl  hei|^ht  >'f  the  w  ater-liarometer,   "i   about 

34  ft. 

(ientrally  spcakiii},',  the  pressure  at  every  pomt  ol  a  con- 
tinuous lluid  must  be  positive.  A  nct^ativc  pressure  is  eiiuiva- 
lent  to  a  tensi<in  wliicii  will  t.Mid  to  break  up  the  continuity 
l)resu|)posed  by  tlie  fornuihe.  Sh«iulil  ne^'ative  pressures  result 
from  the  calculations,  tlie  inference  would  be  that  the  latter 
are  based  ujmhi  iiisulTicient  liyiKitlieses. 

The  pressure  in  water  flowing,'  throiitjli  the  air  cannot  at  any 
point  fall    below   the   atniovpluTic     pressure.      There   are   cases. 


/</-h'\()!7///-s     THfORIM. 


'  "''•   ''y^'  "''V'  ''"  '''•^-^'--"■'-■'''   '-•■-  tins  ln„h  a^ 

tliL  .ur  1,  present  m  snitu  ,cnt  \,,liin,c 

„      ''T''''"   '   '^'"^"'   '"■•'   ^'""^    '"    ^'^-■»'>'   l-nndcil.v   two 
T"     T""   ''^-■-  •^•  -^^  --'Mot   :,.  .Jh.  tlK.-,,u.an 

principle  of  coiit:niiit\. 

and  the  velocities  are  i.uepM.Iy  as  the  sectional  area. 

A^a.n,  assun.e   that  a    ,nov,n^   mass  of  ,h,i<I   eons,sts  of    .„ 

-"Ujn,,,„H,r   of  strean,-hne.,.,u,eo„.,er  a    portion  oft 
nasshoMn,!,  ^tream-lines  and  l.y  tu„   pL.nes  of  ,.re  ..    / 

■', -It  n-ht  ,     .,K,  to   the   .hVection    of  tl.,,v.       J, 
mean  velocities  of  i!ou   auo-,.  d,,.  pf,,,,.,^ 

'■'■''  =   ■-'-  ■"■.••'  'I"''i'-  ''"id  1^  nu-onipres^ihle. 

A-umw,,,  'l"!the,ln,d,.  .o,„pre.s,h]o.andth,a,l 
sp^ancw.,,,,       .Uu.t.o  planes  are   :e,   and...    then';^^^ 
^^.,oni.,     Houu,,  across., ,,se,,ual   to   the.. :v,,,,,Hch 


•■  .  are   the 


IK'  mean 


flows  across  ./,.  siiu,  ti,,.  „, 
remains  constant       |  K  nte 


I.-; lit  ..f  tluul  /'.-Aw,/.,  thetuopl,,,,,.. 


I-    liased  on    tile 


4-  Bernouilli's  Theorem.     "I  hi.  tluore,, 
tollowin.;  assumptions  : 

^.'^    ■'■''■"    ""    """'    "^' !-■   -.nsi.ieration  is   a   ./<W//r 

moving,  stream  mad,  „p,„  ..„  infinite  number  of  .tream-Iinos 
Whose  paths  m  sp,,ee  are  nccessaril.v  (l.ve.l 

(2)  That  tlie  veh.citie.  of  consecutive  s.ream-lines  an-  „ot 
Hidely  different.  s„  that  vi.scosity,  or  the  frictio-al  resistance 
between  the  stream-hnes.  is  sufficiently  sm.dl  ,o  |,e  clisrc-anlccl 


BEHKOUiUrs    THEOKhM.  9 

(3)  That  the  tluid  is  incompressible,  so  that  there  can  be 
no  nit,  Ilia/  -^'ork  due  to  a  clian};e  oi'  vohmie. 

In  any  t^'iven  stream-Une  let  a  portion  A/!.  V\g  1,  of  the 
:lijid  move  into  the  p.-sition  J  /;   jr,  /  .-,cconds. 


Ku;.    1 


I-et  ,?,  .  /»,  ,  :•,  .  ,r,  h<-  tlif  normal  seLlion.il  area,  the  intensity 
of  the  i)ressiire,  th.  vd.u  ity  ..f  |l,,u,  .iti.l  the  elevation  above 
a  (latum  plane  ,:,.-  ,,1  the  thml  at  J.  Let  ,-,,/,,  r, ,  .-,  denote 
similar  (juaniities  at  J!. 

Sinee  thr  intmial  \\,<\\  i-.  ni!,  tin-  \\,.|-k  d^nf  l)v,i/,r;/,?/ 
forces  must  he  cq.ju.di  lit  t..  till'  eiian-e  .>!"  kmetic  enerjjy. 

V    ,,   ,1,  .  I  >  ^''^'  ^^"'■'<  <lonr  h\-  "ravitv 

I  4-  tlu'  Work  d  ine  hy  ]»ressure. 

I^ut  ulun  tlu-  lhii,l  .;/;  parses  int..  the  poMti,>n  .  /  /;  ,  the 
work  done  In  -ravitx-  1-.  <■,  pir  alent  t..  the  u,.,k  doru-  in  tile 
transference  of  tlu  portion  AY,',  and  therefore,  /  bein-  the 
time. 

t/it   x.-ork  ,{oiu  i'y  };);rxity  =  aw,  .  A  A',  j,  —  wn, .  BIV .  - 

=  TiW,  .  ?■/  .  c,  —  lua.^ .  T^t  .  ~j 

since  .lA'       7/,  /.7f  :=  7/.  and  ,/,:■,  =  C'  -  n.v.,. 

Ayain,  t/if  zcori'  hue  by  tliv  {<ressur,s  on  the  ends  .-/  ,,nd  /,' 

The    work    done    1)\    tlu    pressure   on    the    surface    of   the 


bhKSOl  ILII  S    THEORRM. 


SI 


stream-line   l">et\M,Tii  .1    aim  A'  i^^   nil,  since   the   [)rjssure   is  at 

ever)-  point  normal  to  the  direction  ofmotifji; 
T/ic  c/uiKge  oj  kiiiilh-  ni.ii^y 
—  kinetic  em  rL;y  of./'/)    —  kinetic  enerijy  ot  .-l/y 
=  kinetic  enert^)'  c)f />/>'    —  kinetic  cner^'\'  of././', 

since  the  motion  is  stcaiiy,  and  tlure  i^  tjicrefore  no  cliani^e  in 

tile  kinetic  enerc;)'  ol  the  inti.'rmediate  poition  .]'/>'.      Tiius 

///<■  c/taiii^c  of  kiiiitii  iiuri^y  =      (?,/)/>'' '-"    —  '  (?,././'- 

IV  v.?         7.'  TV 

=      ir.,vj    ■  —     <;,:•,/    ' 


Hence,    equating;    the    external    work    and    the    chan^^e    of 
kinetic  eneri;y, 

"'(?/(-, -^:» +(?/</,  -A>= Jc/('!:'  - '/), 

which  ma\'  he  writti-n  m  the  torm 


+  /,  + 


-<■-..  +  /.  + 


or 


f,  J.  S  A  -.,2 

7<>  2i^r  -       I        J,,       I       2j,r 


(w 


(2) 


But  ./    and  /.'are   arhitraril}-  chosen    points,  and   then  fore, 

at  any  point  ot  a  stream-liiii  ,  the  motion  heinj,'  steadv  and  the 

\'isc()sity   nil,  tlie  f^radn.al    mten  h.inije  o(   the   cneri;;ies,  due   to 

he. id,  pressure,  .mil  \elocit\  ,  is  expressed  li\    the  (.([uation 

w  v' 
W2  -(   p         ^    ^        wH,  a  constant  ;   .      .  (3; 


or 


p        V- 
z  +       -I-        -    H,  a  constant ; 

W  2g 


.     .      (A> 


Z  bfinj,'  the  elevation  ot  the  point  above  the  (latum  plane,  p  the 
j)rcssure  at  tlic  pomt,  W  the  specific  ucij;ht,  and  v  the  velocity 
of  tlow.      lliis  is  Bernouilli's  theorem. 


d] 


^'2) 


r; 


5 

1 


BERKOLIl.llS    THEOREM.  li 

Thus  tlu-  tnt;il  const. mt  energy  of  wH  ft. -lbs.  pei  cubic  foot 
of  tluid,  or  H  ft. -lbs.  per  pouml  of  fluiii,  is  distributed  uniforml\- 
;don^^   :i   stream-line,  wH  beini;   made  up  of  wz  ft. -lbs.  due  to 


lu'ai',    pit. -lbs.  due   tu   pressure 


W  V* 
g   2 


ft. -lbs.  due   to   \cluc't\-. 


and  H  beini^   made   up  of  z  ft.-ll)s.  due  to  liead,        ft. -lbs.  due 


w 


to  pressure,  and 


2g 


ft. -lbs.  due  t(j  velocity. 


Hence  the  total  eiier^^y  is  made  up  of  three  elements,  and 
each  element  m.iy  be  utili/ed  b\-  ,i  specially  designed  motor. 
The  now  almost  (ibsolete  oxershot-wheel  is  dri\en  by  the 
Tivv;'-/'/  of  the  water  filling,'  the  but  kets  on  one  side  and  descend- 
ing.: from  .1  hi^'her  to  a  hiwer  le\el.  In  the  breast-wheel  and 
cert.iin  turbines,  the  energies,  due  both  to  tlu'  r.'iV;,'-/^/  (:,'._-)  ami 

to   the  ~,cloiity  I  1,  .ire  traiistormed    mto   ust'tul  vvork.       The 


■:,<-.■■'■ 


rot.ition  of    impulsi -wheels    is  dm-   to   the   kii::ticiiii-rc:y\-'^'— 

^  2ir 

<if  ,1  jet   of  uatiT   issuiuL:   from   a   nozzle   and   im[iin^Mn^'   u[)on 

cur\(d  buc'et-.       I'ln.dlw  tin-  pistiiU  of  tlie  h>ilraulic  en^'iue  is 

attu.ited    b\'  water    admitted    mto    the   c\  luider   from    ,i    closed 

]ii]H'    \\\    which    the    w.iter    under   pressiue    mo\'es    with    ,i   low 

velocitw 

As^-umiiiL;  tint 

((?  I  the  motion  is  stead}'. 

(/')  the  friction. d  resist. nice  m,i\'  be  d^n  Lj.irdeil, 

I'  )  the  fluid  IS  incompressible. 
Kernouilirs  theorem  may  be  ai)])lied  to  currents  of  llnite  si/c 
at  an_\-  normal  section,  it  tlie  stre.im-lires  across  that  section 
are  '-ensilijy  rectiime.ir  .nul  ]i.ir.dlel.  Then'  is  tlien  no  mti  rior 
work  due  to  a  clian^,'e  of  volume,  and  tlie  distribution  of  the- 
lui'ssure  ill  tlie  section  under  i  onsideratioti  will  be  the  same  as 
It  till  lliiid  Were  ,it  re--t.  th.it  is.  in  .iccord.mce  witii  tiie  hvdro- 
static    l.iw.      This    is    also    true   wheth.er   the   llo\\    t.ikes    place 


I 


12 


■4r Pile. -moss  oi-  lirRsornirs   ihijwfm. 


inuicr  atnio.ph.ru  prc..urc  nnly,  or  u  hcthcr  tlu- iIumI  „  ul„,lly 
or  partKilly  cnntliud  l.y  ,,,lul  boun.laru..,  as  ,n  p,i,..s  ,,nd 
^•■•mal.,  or  ulu-tluT  tnr  ,l,,u  ,.  thn.u.^l,  another  nu.liun,  ..heady 
•■^^npud  h\  a  x,,iun,.  .,f  the  tlui.l  at  rest  or  ,„,.v,n  ■  >tcad,lv  ,n 
^  paralhl  .hrcefon.  1„  the  last  eas.  there  nu,  .t  n,vessanly 
be  a  Lateral  e..nneetion  l.etu.en  the  tuoilui.ls,  but  tin-  pressure 
"ver  the  seet,o„  „u,.t  UA]n^^  tlie  hydrostatie  Kiu  thnn,..hout  the 
separate  llnuU,  and  there  ean  he  no  sudden  el,,n„e  of  pressure 
at  the  surlaee  of  separati,  ,n.  as  tins  w„uld  lead  to  an  intcrrun- 
tion  ot  the  I ontinuitv  . 

riu'  hxpotluM-,  JiMuex.T.  upon  which  these-  result,  ux" 
i-ase.l,  are  ne^  ,t  .•x,u  tl)  reah/ed  ,n  aetual  -■xpenencc.  and  the 
rcMilts  ean  only  he  re,L;ar<ied  as  tentative,  l-'tuther,  thcv  ean 
""ly  apply  to  an  indetinitcl>-  ,h,,rt  h.v^th  of  the  eurrent.  as  the 
Vi>ros,ty,  ulneh  ,s  p-,,p,,rtional  to  the  surtaee  of  eontaet.  uould 
otjieruise  become  to.,  ;^r,-,.,,t  to  he  disregarded, 

N  5-   Applications  of  Bernouilli's  Theorem.— n  a  das-,  tube 

*'lHn  at  both  ends,  a.,.1  called  a  pie/onuter  (,T/ere/r?  ,,,  p,-,.ss'; 

u£T/)ni  ,  a  nieasurei  i>  inserted 
\-erticall>-  in  the  current,  It.;  j, 
at  a  p,.int  .V,  ,;  ft.  ,d>n\e  the  pwuit 
''  in  tile  d.ituni  hn,-.  thr  \\,,t,r 
^^'11  i"'-^''  in  the  tube  to  ,(  hn^^ht 
•'/.Vd.pcp.deiit  upnn  the  i)re-..ure 
at  A".  11, o  .■ffect  of  the  e-d.ly 
motion.  i)roduced  ,a  \  b\  ub.truct- 
u\ii  tile  stream-lines,  ni.iy  be  .Imim- 
islied    by   niakini^r   .],,,   ^.,,,|    ,,,    ,||^. 

f'''"-     IMrallrl    to    thr    direction    of 

How.       Disrc^ardin'^:  ilic    effect    of 

'•'<■   'ddies  and  takin-   /,  and  />   to 

'';;""■   '"""■^'<"-;  '■'   «i"-   p.-.'ssurc  at  .\and  of  the  atmospheric 


!» 


Fir..   2. 


pressure,   respeaivelv  .   tiieii. 


"-y+§. 


.■fi  rnc.-iiioNs  Oh  mrxol'jllis  rHtx)Ri:M. 

111(1  tli'-eforc- 


+  t  =  -  +  .l/.^'+:^" 


=  ().\--u  .I/.\"  + 


A, 


=  ('-'/+ . 


A, 


'3 


(5> 


Tile  locus  of  all  such  points  ,is  J/  i-,  ond,  dcsii^natcd  "  the 
line  o|  hydraulic  -radiciu, ' '  or  tlic  "\irtual  s!,,])c,"  terms  also 
used  -aIuii  friction  is  taken  into  account. 

l.et  tin-  two  piezometers  Ail,  (/'.  I'Il;-.  ^,  Ije  inserted  in 
the  current  at  any  two  points  /,'  and  /'.  ;,  ft.  and  :,  ft. 
res[,ectively  aixive  the  jxiints  /:  and  /    in  the  datum  line. 


Kiu.   3. 


l.et  /,  he  tlu-  inti  nsit\  of  the  pr<.sure  at  />'  in  pounds  per 
sijuarc  foot,  />_,  that  at  P.  and  let  the  water  rise  in  tliesc  tube.s 
to  the  liei.i;hts  A'./,  /)('.       Ihen, 


A, 


-f  .;/:  = 


^\      and       ^"   4-  il-  ~   ~   _.    ll 


sM 


M  APPLICATIOS^    Ol-    HLRSOLILLIS    THhX)RFM. 

and  therefore 


■f>) 


tin-  line  .Ui  beiiit;  parallel  to  the  datuni  line. 

'■'"'-•   (-.+{'';)  -  k.  +  ^;)  is  e.jual    to  the  fall  of  the  free 
-surface  K'vel  between  the  poi!U>  /.'  and  /). 

Let  ,■■,  .  .-•,  hi    tlie  \eloLities  of  tlou  at  /.'  and  /'.       Then,   by 
Hernoiiilli's  theorem. 

'  -^  '^'  4-  '■'"  -  ■   a-  ^^  a.  '■-•' 
and  therefore  the  fail  ol  free  surfue  ie\el  !)i't\\ceii  //  ,ind  /» 


•I 


=  I- +;;;-(-+;; 


■-(Illation  (71  nia\-  .dso  be  written  in  the  f)rin 

/S::n-f^,+^.')-(..+^:)=-C+(v;. 

-.-.  -.s  '1-  /  \  11'/  2i^ 


{^) 


so  that  the  vehxity  at  /'  is  e.ju.d  to  tint  .iciuire.l  b\  .1  body 
with  an  initial  veloeit\-  :■.  fdhn-  treelv  throuo],  the  v-rtiea! 
tlistance  Cd. 

Iroiiile  iliu.tr.ited  I  !i  i  iiouilli's  theoieni  I'vperinient.d  1\  i)\- 
means  ..fa  tub,' of  \  arym.L;  sectio,,,  !-,o.  4,  eon\e>iny  .,  current 
between  two  eisterns.  The  i)ressuie  atdiff,  rent  p,  ,in!-,  .dont,^ 
tlie  tube  i-^  measured  b\  pie/onieteis,  .md  it  k  fuind  tii.ti  (he 
water  st.mds  hi!;h,  r  .md  the  [.ressure  is  theref.re  -le.iter, 
wlicre  tliecros.s-section  is  l,,r-er  an.i  the  velocity  coiise?|,icnt!y 
loss.  Reynolds  illustrates  the  prinLi|)le.  th.it  the  pre-..i„c  ,,1  a 
frictionless  i)ipe  ,,f  varyitijr  section  increases  .md  diminishes 
witli  the  secti'Hi.  by  foreini;  water  at  a  \\\^h  velocity  throu^^h  .1 
^in.  pii)e  drawn  down  m  the  nnddle  to  a  bore  of  .05  inch.      .\t 


.irri  h..iTi')ss  or  hi-:r\()i  ti.i.is   ihia)ki:m. 


y(>) 


fret 


(7t 


(>^) 


I 


1 

5 


this  |)Miin  tin-  i)ics-,ur(,-  i^  so  iiuk!;  <linii!ii>lu(l,  tliat  tlu'  uatir 
hwscs  and  bdil-..  'I'm  thr  -,anu:  raii^i-  i^  ciuc  the  hissiiii;  sninul 
licai-'!  ill  \\ati  r-jnj(.'i.tnrs  and  ill  partial])  (.piMUil  \alvcs.  If  the 
sccti<.ii  of  tile  throat  at  ./  i-,  sucli.  tli.it  tiic  ve!<icit>-  is  that 
acquired  b\-  a  bod\-  faUiiiy  freely  tlirouj^li   tiie  vertical  distance 


/;■  hrtuicii  .;  and  the  surfu  c  K\xl  of  tii.  \\-.(ter  in  the  cistern, 
and  il  j'  >ic  the  pre^-ure  ,tt  ./,  aiul  ;  the  elevation  of  .  i  above 
datum,  then,   nei^leetnii^Mric  tion, 

lint  ;•'  =  2.-/.  and  tlierefore  /<  /.,.  so  that  tile  pressure 
at  ,1  IS  that  due  to  atnio,plicrie  pres-<ure  only.  Ihus,  a 
jvirtion  of  the  pipe  ni  the  neighborhood  of  .  /  max  1h'  removed, 
as  in  (he  thro, It  ot  the  inji-etor. 

.\-,iin,  let  the  cross-section  in  tlie  tliro.it  ,it  /.'  be  less  than 
that  at  ,/.  Idle  pressure  .it  /.'  will  Ix-  le-,s  th.ui  the  .itmos- 
pheric  pressure,  and  a  column  of  water  will  he  lifted  up  in  the 
>  "rved  ])ie/ometer  to  ,i  heiyjht  //' . 

'•'-■^  ";■  ■;•  .•'^,  •  '.  '"■  '!"■  section,,!  m\-a,  elevation  abocc 
datum,   pressure',   and  \  elocit}-  at   />'. 

I.et  a.,  .:,.  /•..   :  ,  l)i    similar  .Muibols  at  /;'. 


i6 


.^pruc.rnoxs  n,-  iu-rsoi  luis   THtOKEM. 


I'Ir-i 


<i 


V^ 


/. 


-  +  ;:+:;-^, +  ^'-^^--  •^■' 


[9) 


''"'^^^=  -  +  a.-  ^''"  ''--'^''^  ■•'"-  ■l..tn,n  ,„  uhuh  tlu: 
^vau,-  ,.  ..hs.rvc-,l,„ns.  „Mlu.  Pkvouk.Ut  ,us..,t.,l  ,u  /:.  ,.ul 
als,,  let   //,   -  ,. 


/'■■_/, 


//,  -  //,  = 


hen 


since  „^;.,  ==  „^,.  ,  „^  1,^.,,,^  ji^^.  ,^.^.,i,,,,.^,  ,^^.^  ^^  ^^^  ^, 


Ilrrcforc 


'',-  —  ir- 


//.  -  //, 


a.>    ..,n.,i,,n  in,    .,H.    M....W    vd,.,ty    ,,r    n,.^     ^^ 


'',''.. 


„._,.«^.v /A -//,,. 


(  lO) 


I  ills  i-  •••!<■    principle   ,,r  th.     .,. 


l"'-.it.ir  ;iii,l    ,iNm   ,,(•  tli,^   \-,.nt.,r; 
^^itlim   A   per  eeiit. 
,.      "'^•7""'   'l-"t,tv.,rn„u   is   fonn.i    bvmnlt.plvnv.  e,,n. 

,     .  ';""""•■    ■"    ^-    ''^    ■"-"'■^■'■.    tl'^n    /A   is    merelv'ti.e 

piezometer  ni<erte,l  ,it  /:  "iin,it\ 

-\;^.iin.   I-ri(ii.l,     ,iK,.    points   ,,.it    tli  ,t  ,.i 
^       1  ■       ■  '  '■''  \^''cii   .un-  niinil)(T  .>r 

tlK   pressures  on  tlK.  cnnur^n,,  ,„,,!  .i,ver.,n,  p,,r,.,,ns   ,i  !,; 


F.\.-i.vri.[-:s. 


',9; 


17 


pipe  will    balatur   racii  nthrr  if  the   sectional    arra.  and  din-c- 
ti'ins  of  the  eniK  ,uc  tlie  same. 

Kx    1.   Ono  CMbi,    f,,„t  of  vvafr  ,„.,•  serond  flows   steadily  throiK-h  a 

fnctiuniess  pipe.     Al  a  point  ../,   .00  It.  .bove  datum,  the  scct.onal^irea 

..t  the  p.p..  IS  .12,  s,,.  ft.,  and   Uie  pressure  is  2500  lbs.  per  sq.  ft       Find 

tne  total  eruM-v  at .-/  per  rub.c  foot  of  water.     .At  a  po.  u  A'  ,n  the  datum 

ine   the   pressure   is    1250   lbs.   per   sq.   ft.    and   the  sertioual   area    o^.^j 

>'\-  tt.     I-  nul  the  loss  o!  energy  per  rubir  foot  of  water  between  ./  and  li. 

The  velocity  of  ll,,>v  at  ./  --—  =  ■&  ft.  per  sec. 

The  total  energy  at  ./  per  cubic  f  .ot  .,f  Aater 


:^oo         8» 


The  veiocitv  of  n,,w  at  /.' =  -   '       =  iC  ft    ue- sec 

.'.025  I    •  -     • 

The  total  energy  at  A'  ptr  .  uhic  foot  of  water 


I  lO) 


--  o  + 


1250 

f>2.t 


'•4 


24  ft. -lbs. 


re  the   !)ii)c 

100  '    ' 


Hence,  the  loss  ,.1  energy  between  ./  anil  /!  per  cubic  foot  of  water 
=   141  -  24  -  II-  ft. -lbs. 

Kx.  2  A  horizontal  frictionless  pipe,  m  which  tlie  pres.sure  is  100  lbs 
per  square  inch,  gradually  contracts  to  a  tliroat  of  one  tenth  of  the 
diameter  and  then  again  gra.luailv  enlarges  to  a  pipe  of  uniform  diame- 
ter.     \\  hat  will  be  the  nia.yimum  velocitv  of  flow  at  the  throat  = 

The  velocity  al  the  throat  wil!   be  greatest  when  the  pressure  there 
is  ml.      Hence,    if   ;•   is  the   thmat   velocity  ami   iherefo 
velocity, 

_ioo  >'    144         I  I J 

and  ?■  =  I2r.437  ft.  per  sec. 

6.  Rotation  of  a  Fluid.— In  any  stream-line  moving  freely 
in  sjucc.  let  .;/>■(/'  he  an  clement  of  mass  „i  and  normal 
thickness  ,/;;i=  /.V  t  Jt  is  acted  upon  by  the  presstires  on 
.-//>and  AT,  a  1  uue  of  intensity/-  ,.n  the  area  .]/!(-  ,,). 
a  pressure  of  intens:t\-  /-  -f  ,//.  ,„,   the   area  CV>,   its  \veit,'ht  »ir 


00  J  64 


'*  KOT^riON   OF  tLilDS. 

inclined  ;,t  an  .in-lc  a  t..  the   nnnn.d,  and  the 
>'/^,  ;•  beiny  the  radius  ..f  curvature. 


centrifugal  force 


Fig.  5. 


or 


or 


Resolving  alonp  the  normal. 

<1  .    ,//>    —    )>i ^       _     „/,r    COS    «•=     O. 

</  .  ,//>  =  ,«('■'  -|.  ,.  COS  «)  =    "''^ -J^  f-  -^    .  -    .     \ 


thui, 


If  the  stream-line  is   i„   a  h..ri/ontal   nhi 


plane,  «  r=:  90^  and 


dp  _  w  V- 

dn  ~  g    r  ■ 

i<ut  by  equation  (4).  Art.  4.  since  s  is  no 

dH^    ,     .Ip        V    ,lv       v;v       A^\        2v     ,  V        ,lv. 
dn       w    d„       ^    dn       g'r-^dn    ^  ,'     .  r    *   dn  " 


w  constant. 


TCC 


IVHIKIING   I  Li  IDS. 


1 1 V       dv 


19 


The    cxpicssi,,.,    ^y^  -l-^^j    is    dcsi-natci     the    average 
angular  velocity,  or  /A,  ii>tatioii  of  tlu  fluid. 

Atrain.  if  the  stream-Hue  is   horizontal  an.!  i-  ako  circular, 
dn       dr.  ami 

dp      w  V 
dr  "  g  r  ' 

a  differential  ecjuation  connecting'  the  pressure  and  the  velocity. 
If  V  is  a  knoun  function  of  r,  the  ])ressiire  can  be  at  once 
deterinineii. 

7.   Whirling  Fluids.- -Let  a  fluid  mass  whirl   like  a  rijjid 
body  about  a  vertical  axis   I)-,  with  an  an^[iilar  velocity  m^ 
Consider  the   relative  eiiiiilibriuni  of  an  element  of  mass  „; 


i 

J 


Klii.   0. 

at  /'  distant  .1  horizontally  from  the  axis  and  i  vertically  fro- 
the  ori^^m  ('  in  IT. 

Take  /'.I  horizontally  to  represent  th.-  .eninh,^.,l  fop  e 
'//<•..  and  /v.-  vertically  to  represent  the  wei^-ht  wc  The 
reinaminK'  forces  nuist  be  equal  and  opposiu-  to  the  resultant  of 
these  two  forces,  viz  .  the  <hauonal  /'C  of  the  parallelogram 
-■//>'.       1  he  ma},Miifude  of  this  re^nltatit  is 

PC  =  ♦  rwcjr^.  (w.^.r).  =  ,„jr^ ,  ^  '"'^\ 


20 


LEyEL  SURhACHS. 


Its  sl(:po,  ,t,  is  ^'ivcn  by 


Inteyratinj;. 


tan  i<(  -^ 


iiiio'x 


_jr_ 


/ 


c-  bcin-  ,1  constant  <.f  integration. 

Thus  an  in.mitc  number  of  Io,.aritI,mic  curves  can  be  drwn 

-h  tlKU  the  tangent  at   any  po,nt   „,  any  one  of  the  cu^Tis 
.n  the  dn-ccfon  of  the  resultant  force  at  that  p,.,nt       T     s! 
curves  are  called  /,...  .^^,,,,  ,„,  ,,^  ^^,^^^^^J  cuttin, 
--^^r  ..rce  orthogonally  are  designated  ...V  or  <,.  >  .::;^ 

If  ^yi.  the  slope  of  a  level  surface,  then 


+  ;$".  =  tan  /y=  cot  a  ='"^"'-^ 


Integm'ing, 


'".i"' 


r.>'.r 


J  = 


^JT 


+  c. 


c  being  a  constant  of  mtet;ration. 

Thus  the  level  surfaces  are  r<^,-af,oloi,is  of  rev„lnt.on 

'•'^•-  ^'^y  ^7  ^'":/-'-  tlus  result  is   obtained   n.,,,    .,n,plv  ,s 

follows:    Iheflu,deIe,ne„tof„,,..   ...  i„  the    f,ee    .....ue     t/ 


Fir,.   7. 
is  kept  in  relative  equilibrinni  by  (,; 
('^)its  u.ijrht  wv.  and  ^g  the  (lu,d  u, 


'lu-  centrifugal  force  »;.-.' 
■■■-vnrcs.  which  !i!i 


cces- 


FX/IMPLES.  2 1 

saril_\-  c  a  resultant  normal  to  tlu-  tree  surface  at  /'.  br.iw- 
ini,'  the  liorizonta!  /'A'and  the  normal  J'(,  tn  meet  tlie  axis  of 
rotation    in    .V  and    (/,  /'.\V/ is  evi(!eiiti>   a   trian-ic   offerees. 


ant!  therefore 


.W, 


and  ay;  =  ^ 


a  constant. 


Thus,  the  suh-normal  is  constant,  an.l  the  free  surface  must  he 
a  paraboloid  uitii  its  vertex  at  the  pomt  O  ulicre  the  fret 
surface  cuts  the  axis  of  rotation. 

Ex.  1.  Deduce  the  hiw  of  pressure  variation  (,;)  for  water  in  a  ves4el 
moving  slowly  towards  a  hole  in  the  cetitre,  the  stream-lines  beini;  ai.- 
proximately  horizontal  circles  and  the  velocity  of  any  fluid  particle 
inversely  as  its  distance  from  the  axis  (b)  for  water  rotatinR  as  a  ricid 
Ix.dv  about  an  axis  (as  in  a  full  ccntrifu>;al  pump  before  delivery  cm- 
m.-nces).  the  velocity  of  any  fluid  particle  be.ng  directly  proportional  lu 
Its  distance  from  the  axis. 

(.1)  Take  v  =  " .  tlun 


Therefore 


{b)  Take   v  =  br.  then 
I  dp 


2f' 


=  —b*r. 


Therefore 


t  =  ^^ 


L.y  ;.  A  cyhndncal  vessel.  10  ft.  in  hei>.ht  and  i  ft.  in  diam..  is  half 
lull  of  water.  Find  the  number  of  revolutions  per  minute  whi.  h  the 
ves.sei  must  make  so  that  the  water  in.iv  just  reach  the  top  the  axis  of 
revolution  beinj;  coincident  with  („)  the  axis  of  the  vessel  .^  a  L-en- 
eratinj{  line.  ^ 

(.1)  The  free  surface  of  the  water  is  the  par.iholoid  /•('/•  Fie'  S  w,lb 
Its  vertex  at  O,  since  the  vol.  of  the  paraboloid 

=  \  vol.  of  circiiinscnbinn  cylinder, 
=  vol.  of  water  in  vessel. 


Then 


e   ._ 


,.  -  A  O-  = 


latug  rectum 


and 


«k>=a  ^31   X   IJo 


I    /'A" 

ifif'to. 


I 

So' 


22 


sHARi'-inai-n  oruichs. 


The  linear  speed  of  the  run  at  /' ^  'm  =  .S|'i<j. 
and  the  number  of  revols.  per  niiii.    —   —  * — ^-^3  =^  ^S'  96 


r.vp 


I 


Fir,.   =!. 


Fi';.  0. 


{/>)  The  free  surfaie  i-<  now  the  paratjoloirl  ( )/'.  with  its  vertex  at   d, 
g    ^  I   PX'  _     , 

M'  2      (^.V  20' 


Th.-n 


and 


4640  ■-.  Hf^o. 


Thus  the  niitnhcr  ol  revols,  pci   1111:1 


60   .    Si  ii 


.     -•41.4S. 


8.  Orifice  in  a  Thin  Plate.  Ii  .m  ..|)rnmL,'  is  m.i.lc  m  tin- 
wall  or  bottom  of  a  tank  coiitaiiiiiiL;  water,  t!ic  tliiid  p.irticies 
iiiimf(iiattl\-  move  towanU  the  opciiiiio.  .md  arrivi'  there  with 
a  velocity  deju'iidin},'  iipoii  its  ih-pth  Iu-Iom  thi'  tree  suifacc 
The  opening,'  is  termed  an  "orifice  in  a  thin  plati-,""  when  the 
water  sprinj^s  clear  from  the  inner  edfje.  and  escapes  without 
aj^'ain  touchin}.;  the  sides  of  the  orifice.  This  occurs  when  the 
boimdin^'  surface  is  chantjed  to  a  s/iar/'  ci/xf,  as  in  V\^.  10, 
and  also  when  the  ratio  of  the  thickness  of  the  hounding,'  sur- 
face to  the   leaNt  transverse  tlimensi.m  of  the  orifice,  does  not 


SHJRr-l-.DiaEU   t  )RU'h JiS. 


^3 


<L'xcce(i  a  certain   atiiMniU  wliicli  i--   u-'ii.illy  fixccl  at  iinit\  ,  a-;  in 
I-'i^'>.    I  I  .iikI  I  2. 

()\\in_L;   to  the    inertia   ;K'i|iiirr(l  li\-  tin'  tlinil  tilaiiuiit'^.  tlie-re 
will  he  III)  ^iiddt'H  eliailLje  ill  tlieir  (iireetim!   at  tlie  eil^c    ot'  tli 
iiriticc,  am!  tli<'\    will    cnntimie    t"   eoiivcrj^e   to  a    point    a  little 
in  front   ot   the  ontiee,  where  the  jet  is  ohserxed  to  eontr.ict  to 
the  smallest  .icctioii.      Tliis  portitm  uf  the  jet  is  ealletl  the  rtiia 


V 


/ 

\ 


\ 


Fir.,    in. 


Fk;.  1 1. 


Kie..   I! 


iontrifita ,  or  lotitr.uteii  \eiii.  .md  tin  l1uil  tllanu  ntsil  ou- across 
the  niininuiiii  seetion  in  seiisihly  p.irallel  lines,  so  that  lure,  if 
the  motion  is  stead)  ,    Hernouilli 's  theorem  is  .ipplie.iMe. 

'Ilie  dimensions  of  the  eontiaettd  section  .mil  its  distal. i.  e 
from  the  orit'ice  depend  upon  tin  torm  .vu\  dimensions  ot  the 
■ontki    .md  upon  tiie  lu.nl  of  Water  o^er  the  orifice. 

I. it   li^     I  ^,  represent  the    jvirtion  of  the  jet  hetueeit  a  cir- 
,  ciilai     orifice    ot     dnmeter    .  / /•'   ami    tht      i  ontr.ieted 

-'  section  of  di. ureter  ( '/',  //  hem;.;  tin  di-t.ince 
between  A  li  anil  L  H.  Tlien.  t.ikint,^  tiie  aittxj^^c 
results  of  a  number  of  (ibservations,  it  is  found  that 
Ali,  CD  aiui  /:/•'  are  in  tlic  rntids  of  loo  to  8o  to 
50. 

'^"'    '^  Tims  tile  areas  o(  tlie  contracted  section   .md  of 

the  onliee  .ire  m  the  ratio  of  K)  to  25,  and.  jjen'-rally  speaking;, 
this  is  assumed  to  bo  the  ratio  whatever  may  be  the  form  of 
the  (»rifice. 


M 


*4  TORRHEIII  s    THEOKHM. 

9.  Torricelli's  Theorem.-I.ct  I',-,,  ,4  nprc.cnt  a  jet 
i^Miin-  tn„n  a  thin-platc  nntu  c  ,.,  tlu-  s„lc  ,,f  .,  vessel  CM,Ua.n- 
iiu:  water  kept  at  a  constant  le\ei  ./A' 

I.ct  .\-.\-l,e  the  (latum  line.  .I/.\- the  contracted  section,  and 
consider   an>st,cani-l,ne    wy/,  w  beni-   in    a   re-ion  where   the 


Ki.;. 


;.   14. 


^clnc.ty    is    sensibly    .e-o.    an.l    /.    in    the    cmtracted    sect..,n 
Ihen,  by  Hernomlli's  theorem,  the  mnti,,n  heinu;  -te.uh-. 


A    ,    '-V^ 


^.  + :.,  + 


/- 


2,^ 


+   -+2.C-        • 


•       II) 


/.  /,  1..  in-  the  pressures  M  :i  and  »/.  ,,nd     ,    -    th-ir  eievati..n, 
al)(i\f  datum        t  lencc 


■2X 


ll  the  t1<.u    is  intM  the  .itnii  .sphere 


A  -/ 


(2> 


/  —  the  atmospheric  jiressure  =  />  ,  and 

A  =  r''.  <>"'  +  /„. 

(^b<Mn-    the    p..mt    m  ^.\uA^  th.     verti.,,1  thmu^h    n,   mterscct. 
the  trei-  surface.        I  hus 


2g 


=  z,  -  z  -f  Om  _  h. 


(3) 


h  bein-  the  depth  ..(„  (h.|,,u   the  free  Mirlace. 


1 


Ti^RRICr.l  lis     IHt-.ORFiM. 


25 


The  result  -ivcn  by  r(iiiation  ,;,  ua^  tlrst  cU-diar.l  hy 
'i  Mi-rjcrlli, 

•Ilu-  .Icptli  l,rl,,\v  tin-  ftcc  suil.Kc  N  \ci-y  iu'ar!\-  tin-  same 
('■•r  all  ]),.ii-t->  .,)  tin-  i-.,ntr;utc(l  \rin.  an.l  the  \alue  .  .f  v  as 
Uiven  tyi.^i  IS  taken  t..  be  the  thenretica!  mean  veh  .cit\-  nf 
flow  acniss  the  c  .iitraeted  seetii'ii. 

iMjuatioii    (^1   is  eiiiii\aleiit    tn  the  statement  that  when  the 
orifice   is   oiKii,,!,    the    li><ln  istatic    eiieri,ry   nf  the    w.iter,    \i/., 
h   It. -lbs.  per    poiui.!.  1-,    c.-nverted    int..    the    kinetic   ener-y   ..t 
V- 
2g     It. -lbs     ]Kr    [,,.„n(l.       Tiuis,   it  the   jet  is  .lirecte,!    verticall_\- 

iipwar.ls.  it  will  ver\- nearly  rise  t.,  th<'  le\ei  of  the  free  surface. 
and  \v..ul(l  re.icii  that  level  were  it  n..t  f<.r  air  resistance.  ,,r  t..r 
viscosity,  ,,r  for  fncti..n  a-anist  the  side-.  ..f  the  ..ritlce.  or  f.,r  a 
conibinati<.n  of  tliese  retarding;  cause-,. 

II  the  Jet  issues  in  .an\-  oth.  r  directi..n,  it  describes  a  para- 
bolic arc  ..f  which  the  directrix   liis  in  the  iree  surface. 

Let  (  '/  /  ,  I'l-.  1  ;,  l,c  such  a  jet.  it^  directi.m  at  the  orifice 
at  i'  iiiakin-    an   an-le    „  with  tile  vertical.       With  a   i.n.])erly 


F.,;.    15. 

rorme.l  , .ritlce  a  -reater  or  less  Iriij.tii  ,.f  the  j.  t  will  Ii.k.  th< 
appearance  of  a  },'lass  ro.l.  ,1.1, 1  if  this  p,„ti,,„  uei,  sudd,  i,ly 
-olidified  and  supported  at  the  ,  nds,  ,t  u.,nld  st  in,!  ,1-  an  .uJi 
without  any  shearing'  str<ss  in  n-.rni.d  sea^ns 


i 


36 


I'f-SSHLS   l\   MOTION. 


A-aiii,  tlu-  li,,n/nm,il  o.niiK.ncnt  ,>r  tiu-  velocity  ..f  flnu  at 
any  m.miU  n-"  the  j.^t  ,.  constant  (=  :■  ,„;  ,.,.  s,.  that,  (nr  th.- 
unbroken  p,.rci..n  ..f  tli.-  jrt,  rrnidi.tant  xutical  plane.  w,ll 
intercept  eiiual  anKMiiit-  . if  Neater,  and  the  hei-ht  ..f  tli.'  C.  (i. 
<if  the  jet  above  the  )iMri/.,„tal  hnc-  (U\  uill  be  tuutlur.N  of 
the   hei-ht   'if  thi'  jet. 

10.  Efflux  through  an  Orifice  in  the  Bottom  or  in  the 
Side  of  a  Vessel  in  Motion.— Ifa  vc'ssel  enntamni-  uater  :  ft. 
deep  ascend  ,.,  cK-ce,,,!  vertically  nith  an  acceleratm,,  /,  the 
pre.-Mire  /  at  the  bottom  i-,  -iven  by  the  e.iiiation 


/„  bein}.;-  the  atnio.plu;iic  ])resmire.      Theretore 


D" 


';/■'-■<- :0. 


l^elori-    .111    orifice     is     opened,     if 

^        the  heavier  \csse|  is   reduced  to  rest 

1>\   appl\in-    an  iip,\,ird    acceler.ition 

./•     t'"'      pressure    ,i|     the      depth  is 

ihan-ed     from     .-.■:     to    u' :[i  -{-^), 

while  in   111,-  Mthe.    \essel   it  would   he 

chan-ed  !r,,m  .-.•:■  to  7,':(i   —      ). 

^  "pened  ,it  the  bott..in,  the  '.eloeit}-  of 
•■niux  ,  1.  still  t,,ken  as  bcin-  d,u-  to  the  hea.l  of  the  pressure 
/.  .md  therefore  In-    I'orriceili 's    rheor.in 


Ki>-..  1',, 
It    no"   an    oritlci'   i 


■fill 


V 


Z    I 


f  1 
g 


I  '  t    //■    b(    tne  wci-h 


hi   ot    the  \.-ss,l    .,n,|   \^ater,  .-md  let    the 
vcs.scl    iu     .onnea.d     with    a    coimte-poise    ,.f   wci-ht    /I',    by 


//');/•   /\    fh'IcnONLFSS   /•//'/?. 


27 


iiican-^   nf  a    vnpr    iKissin-   ..vcr   ,1    piilk-N-.        Ih,.,,    by  .New  t-.n 's 
sicirnii  l,n\   of  nwitioii,  ami  itc-lcLtin-   piilKx'  IriLtidii, 


s 


ir. 


//:. 


/'■, "  //'  • 


7  hLJiiL;'  the  tension  rit'tlic-  n.i)r.     riurcforc,  also,  7'=     ~ 


u-jr 


Next  k-t  a  cxlimlrii'al  Nt-s^^cl, 
I'ii;.  17,  of  radius  /■  and  ^ 'intaiinni^' 
water,  rotate  uitli  an  angular  \e- 
locity  (.'about  its  axis,  Art.  -.  'Ihc 
surface  of  tin-  water  assumes  the 
form  ol  a  ji.iraholoid  \\  itii  its  \ertex 
at     (*    .md    its    l.itus    ri'Ctum    ecjual 

'"       ,•       It   an    oritke    is   ni.iilc  at  O 

ii)  the  side  of  the  \t'ssi-l,  at  ,1  \crti-  Fi<-   17. 

cal   distance   ;    from  (>.  the  watci-  will  How   out   with    a    \elocity 

:■  i\w  to  the  llead  of  pressure  at  the  orifice,       'Ihw  head  is  /'(J 

and 


/V>       '',\- 


±  ~. 


the  sit;n  heini; ////.>•  or  ii/i>//fs,  accortlini,'   as  th>    onticc  is  helou 
'•r  .ibove  (■',       Ileiue.   by  'iorri..  elh '<  thenrcni, 


±     ~. 


or 


V*  =  (. -'r'  ±  2gz. 


II.  Application  to  the  Flow  through  a  Frictionless  Pipe 
of  Gradually  Changing  Section  di-.  iSi.^l.et  the  pipe  he 
supplied  from  .1  ui.iss  ,,{  uater  of  whicii  tlie  free  surAice  is  //ft. 
above  datum. 


38 


//.()«-•   /,V   l-HlLIJO\Lf;ss  PIPE. 


1^^^  ",•  ^,.  r,  be  Ihc  scctir.nal  a>ca,  j.rrssuro,  an,!  velocity 
"f  flow  at  .in>  point  ./,  .;,  |t.  abnv,-  .latun'i 
•  iiul  //,  ft.   l,cl,.u   tile  tree  surface. 

l-^■t'^,.  A.  7^  be  similar  symbols  t;,r  a  seen,,,!  ]„,int  A', 
-,,  ft.  abnve  datum  and  //,  (t.  below  the 
tree  surface. 


I 


I 


Fio.    i8. 

Then  by  tlic  condition  of  c(>ntiniiit\-, 


and  by  Torricelli's  tin  <.rc 


=   /',  4- 


/n     -     /, 


and 


Hence 

-r  i 

•    I 

2£^ 


2.1! 


r=^'.+ 


/..  -  A 


A 


+ ;;  +  -,  =  -,  +  //,  +  ^''  =  // + ^ 


/u 


=  -,  +  //,+  ^■•=  •-■  +^^4- - 


HYnh'-ti v/c;  (:nrhhi(:if:\'Ts. 


29 


icity 

ituiri 

t  /;, 

■  the 


so  til, It    lie 


■ 


ni'  'iiil  ii 


u:. 


lu-iilclll,    \I/., 


-  //4- 


/ 


>-"ti-;tant. 


Ii'ild,  true  for  the  ,i^simR(i  CMiidition-,. 

12.  Hydraulic  Coefficients.-    Ihc-c  arc  coefficient,  intn.- 

(liKe.i  t'.  correct  the  (h-^erepaiu  ie>  iietween  the  i-e^ult-  dechieed 
b\-  theoretical  coiisiderati' m--  aixl  th<:  actual  result-,  of  practice. 

Numerous  c.xperum-nts  have  been  made  to  dctoniuiu-  tlie  values  of 
lluse  coetficients,  and  with  the  same  object  in  view,  special  apparatus 
has  been  designed  and  installed  in  the  hydraulic  laboratory  of  McCiill 
University.  A  mam  f,  ature  .,f  this  apparatus  is  a  rast-irr,n  tan!;,  square 
in  section.  28  ft.  m  hri-ht.  .ind  having  a  sectional  area  of  25  sq.  ft. 
Care  has  been  lakm  L.  mak.-  the  mside  surfaces  of  the  tank  perfectly 
flush,  and  to  Ihis  end  the  tlanyes,  by  which  the  several  sections  are 
bolted  totrether,  ;ire  phu  ed  on  the  outside. 

The  valve,  Fij;.  19,  in  the  side  of  the  tank  is  a  gun-metal  disc  ]  in.  in 
thickness  and  24   ins.  in   diameter,  fitted   into  a  recess  of  the  same  di- 


•i«iT  <H.»rt  or  T4H« 


■H^H        !i 


8*C-  0«  1,1s  0£  v<C« 

Fig.  19. 


mensions  m  a  cast-iron  covei-platc.  with  gun-metal  bearing  faces  form- 
ing a  water-tight  joint  for  the  face  of  the  di.sc.  This  cover-piate  is 
bolted  to  ;,n  opening  in  the  front  of  tlie  tank,  and  the  inner  faces  of 
Ihc  cover-plate  and  dis(  .ue  tlu.li  with  the  inner  surface  of  the  tank 


3° 


xmihirjiixr  Of-  u'Hi.oaiY. 


In  tlu.disr.an.ion  ..,,,„.Mt,.  sides  of  th.- ct-ntre.  there  are  two  screwed 

tliL  disc  e.uh   ..,,e,H„>r  ,,,„    be  made  eonccntrie  with   a   screwed  7>.i„ 
openmK   '"  the   body  „f  the  valve.     The  d,sc  is  rotated  bv  means  o>  a 
s,,ndle  throu,.,   us  centre,  passing  through  a  gland  in   the'  front  of  the 
a    e   ,o.ly,  and  operated  by  a  lever  on  the  outside.     Ciun-metal  bushes, 
th  the  reqiared  onhces   are  screwe.l  into  the  <lisc  openings,  and  when 
s  leued  h„me  have  their  ,nr,er  surfaces  flush  with  the  vaive  surface,  and 
therefore   with   the    mside  surlace  of  the  tank.      H,  means  of  a  simple 
Kv.ce.  these   bushes   can   be   readily   re.noved  and  replaced  by  others 
u  thout  the  ioss  „f  more  than  a   pint  -f  wat.-r.     A  cap  with  a  central 
gland  IS  screwed  into  the  r.l-in.  opening  of  the  valve  body  and  forms  a 
practically   water-tight  cover.     The  valve  is  rotated  so  as  to  brhlg  tie 
bush  opposite  the  opening,  and   it  is   then    .inscrewed    by  means  of  a 
special  key  protecting  through  the  cap-gland.     The  valve  is  now  turned 
bark  until  the  opening  is  <do.sed,  when  the  cap  is  unscrewed,  the  bush 
t.iken  out   and  another  put  in  its  place.     The  cap  is  again  screwed  into 
position,  the  vaive  totated  until  the  openings  in  the  disc  and  tank-side 
are  concentric,  when  the  bush  is  screwe.l  home  by  the  key 

A  gu.,-,netal  bush  screwed  into  each  of  the  two  openings  in  the 
...am  disc,  carries  a  series  of  orilice  plates.  The  larger  bush  takes 
plates  with  openm;'s  up  to  4  ins.  in  diameter,  .-.n.!  the  smaller  bush  takes 
piates  with  ope.im..  up  to  .  J  ins.  in  diameter.  The  plates  are  provided 
with  a  shouldered  edge,  whi.h  fits  against  the  corresponding  rim  of  the 
bush,  and  are  screwed  with  the  orifice  in  any  required  position  by  means 
of  an  annul.y  screwe.l  ring  fitting  the  interior  surface  of  the  bushing 
The  or.fi.e  plates  are  gun-n,etal  dis.s,  4!  ms.  m  diameter  bv  \  in.  thick 
f^.r^tlie  larg,-   bn.h,  .uid  2  ins.  in  .liameter  by  i   in.  thick   fo'r  the  small 

Th,.  utmost  ,-are  has  been  t.ik.M,  to  f.,rn,  the  01  dices  with  the  gr<>atcst 
p..ss,ble  accura.:y.  The  orifices  are  w..rke,l  in  the  .ii.scs  appr<..x.matelv 
to  the  sizes  required,  aii.i  are  then  stamped  out  w„h  h.irdened-steel 
pnr,<-hes,  the  s,z,.s  of  which  have  been  detennino.l  with  great  exactness 
by  mean  ;  of  Bn.wn  &  Snarpe  micrometers.  The  .li.mut.-rs  of  the  ori- 
fices arc  also  checke.l  by  a  Rogers'  comparator  ..ii.l  a  standard  scale 
In  some  cas.;s  a  discrepancy  has  been  foun.l  between  the  sizes  of  thJ 
-he  and  its  orih.e.  but  the  size  obtaine.l  for  the  ..nfice  is  the  one  which 
li.is  b.'i'u  invariably  used  in  the  c.il(iilati<uis. 


(..•'1    ('''ftfhlfiit.'!    ,<f 
\'''iia     Cdiitr.ict.i     is    ,1 
v.l.icit)-   (.Art.  .)!.    .iiH 

CMcftujl'Ilt    iif   \cl'.cit\  , 


/  t  IdiitV     - 

little       Icsv 

th.     r.iti. 

I  'clli  itill^ 


riu'  actii;il  \cl.n.-it>-  ,-•  ;it  tlio 
than  ♦  J;'//,  the  tiic.rcticil 
"I  •■  to  \  J,.//  is  cailcd  the 
this  cocrficic'iit  by  <,, ,  then, 


c(  )h.hhi(:iiis  7  ( )/   ;  -HI.  oa  r  r. 


3» 


and  the  rqii.itions  for  tl.r  \cl. .cities   .,f  (li-^cliai-^c  in  the  ca>c  < 4 
iim'Viii.l:  vc'^scls  (.Art.    lO)  licei-nic 


-.,« 


'■,.'•  21  A-"  ±  /)/' 


and 


=  <;-i  :.■■>■  ±  2,i,'-::). 


A  niiiDi  \ahie  i.l  ,  tnr-wclj  fDrincd  sim])lc  orifices  is  .974. 
Assiimm-  tliat  tlie  face  of  tin  onfice  is  vertical  antl  th.it  tin 
jet  issues  hurizontally  with  a  \  eh  .city  of  r  ft.  pci-  second,  nmh  r 
a  head   of  h  ft.   of  water,  and   assnmiiv.^^   also   that  in  /  sees.,  a 


Kio.   20. 

Ouid  jiarticle  reaclu-s  ,1  ix.iiit  r  ft.  nie, inured  verticall>-  .->nd  .r  ft. 
measured  hnri/( >ntall_\-  from  the  point  of  ilischarf,'e,  leii,  dis- 
rei,'ar<hii,i,'  the  elfect  of  air  resist.mce  anil  (jflier  disturhiii"- 
caiiscs, 

.r  =  VI, 

y  =  ic/-'. 

and  tlieref<ire 


1* 

y 


=    ~''v~K^'   -  4'.,-'//, 


C;  =z 


4/n' 


J 


32  coiji i(.,h:\  I  Or  r/i.ocnY. 

•'    'i  •  ■':  ■"■•■  ''"-■   ^  "•i-um.ilo  of  the   tluul   particle  in  an_\-  ntluT 
p'l-itinn,  tlKii,  also, 


I  loiu 


/ 


c.-' 


4//f 


4h(y,  -  y)' 


Mliicl.  i^  the  fnrniiila  iwrd  in  tlir  Mo', ,11  ial.orat.,r>-  in  tllr 
I  ■^l>■■Illn^ntal  .K  tcrininatiuti  .  ,t  ..ulh,  irnt,  ..I  vclo.it)-,  'I  he 
P'.sitiMn  .,t  J,,.  „.,  is.JHinc.l  I.N  xntuai  nira.Miri-nic-t's  (mm  a 
straiiriit-.d-r.  --nppnitr,!  h..,  i/,  ,nt, illy  ahn^.  the  jet.  .j/ a  l.raJ..! 
on  the  tank  face  at  .mk-  en.l,  an,!  at  the  nth.r  ..n  a  iHaiin.;. 
u-liich  admits  of  a  \ertical  .idjiistnu  nf .  ]-l<:,  ji. 


Kr...   21. 


The  strai-ht-e<lj,'c  is  ot  niachmn  \-m<'.  1,  i.  :;  ^  ft  ,,,  1,  ,,.^(h, 
2^.  ins.  in  depth,  ^  in.  in  ui.lth.  .,n,l  ,s  n.aduated  s.,  as  u.^ivi- 
the  horizontal  distances  |,,,ni  t;„  nuur  face  ,,f  the  orifice  pl.ite. 
The  \ertu,(l  n,,l, nates  ,„v  m.asun,!  l,v  .,  W.mier  caliper 
specially  adapted  f..r  th.' piirp..se.  Th.  Hal  fair,,!  th,  niM^.d.l, 
limb  is  made  to  r.  st  upun  the  upp.r  surface  n(  the  strai-ht- 
ccinc,  and  the  c.iliper-arm  han^^s  v.rticallv  .\  Lent  piec'^e  ..f 
wile,  uitii  a  lu.dlr-pnint,  iselamped    t..   th,     .,th.i    linil.,  and. 


3? 


C(J/;7/-7f,7/;,v7'  I  if  yh:iA)(:iTY. 


n 


by  nuMii-  (if  tlu'  -iicw  .uiiustimnt,  i:,\n  Ik-  rr,uli!v  moved  mitil 
it  iu^t  li.iklR-.  the  ujipci-  nr  Inwtr  <urt",n.i-  of  the  jet. 

r>\  m.  ail-^  nf  till'  .ilxivt'  nietiinil,  .ui  extended  -mes  nf 
.  \l>eiiiiient>  with  j  in  ,  l-in,,aiid  i-m.  sh,ir[i-(  d-e  i  )ritiee>,  and 
uiulcr  licads  \ar\-in^'  h.im  (>  i,,  jo  ft.,  jt,i\c  jyj  as  the  a\era;,;c 
value  lit  tile  coel'tk  lent  of  \-eli)eit\-     i .  i. 

Let  tile  dirntiiiii  of  tlie  jet,  V'v^.  2:.  at  tile  puint  nf  dis- 
.  Iiari^c    make   au    aii^le    ,t  witii    the    h.  in/i  iiitai,  and  let  ,r,  .   /•  , 


Kii.. 


i\.    v.,    he    the    ciM.rilinates    ilefinin;,'    the    picition    ul    a    lliiid 
{Kirticie  alter  ii-.tervaN  ol'/,  sees,  ami  /,  s.-c<<.      Then 


(I  .  /,        ,ind 


.1 ,  —  .    ^111  <«  .  /,         ic'','. 


^'«  =  !'  COS  It  .  /.         atul 


1  hesi-  e([tl.ltioiiN  '^\w 


r,  =  :•  sill  a  .  /j  —  i.<,^//. 


•i'l',  -   'i^i 


tan  tt  =  -'''- 

r,4-,(x,  _  4-,) 


ami 


4, •  sec'  rr 


4-,  tan  <tf  -  I',  x 

h-iiui  uhitli   It  .md   then  ..   i  m  Ix    i  ak  ulati-d 


« ,  tan  a  —  j'^' 


34        -joi  iiii:ii;\rs  or  av.-.s/.s/  /vc/:  .^.W)  comraction. 

\/>)  Ccitjiiuiit  ,'f  /\is!sr<iih\\—\.ijt  Ii,.\>r  til.'  ad  nijiiircd 
to  prnilinx  t!  •  VL-locity  ,■■.  I.rt  /',  he  the  lu.  inquired  to 
•  ivxTomu'  tin-  irictioiial  ii  >i-.taiK  <■.       Tlicii 

/i,  tlic  tdtal  luMil,   =  //.  4-  /^  ^  /i^^i   _^  ,.^^^ 

w  Ikti   //,   =  (■,//. 

i,  \^  tcniird  the  in.t'lKi.nt  ■  if  n  <i\taiici',  ami  i-,  apnrn\i- 
niatrlv  constaiu  t(.i-  \.ir\n;,-;  li.a.ls  uuh  MMiplr  -harp-nl-ud 
(jiilkts.      Airain, 


Hence 


and  tlii'rcfdri' 


<V'^  ^     ,       =  'i... 


h  =  <v'/J(i  +.V). 


so  tliat  <•,  can  be  iniind  w  lu'n  r ,.  i-  knnun,  and  .•,, ,    .,  ;.v,/ 

((■)    I  ocJfiiiiHt  of  L  ontiilitloii.        I  lu     laiiu   (.1    tlle    area    ,/   of 

the  vena   d-ntracta   t<>  the   area  ./    <<\   tin    ..rifu.    is   ealhd   the 

coefficient  of  contraction,  an<    nia\   hi    d(  n utid  IjV  < 

The   value   of  <    mii-t    In    deterinnnd    in   each  case.  I.nt   m 

sli.i'l.-. (1^.(1  ..nfices  an  avera},'e  \ahie  of,    .  ;,s  alrea<ly  p.,)!-,:,  d 
I'. 

out.  "^    ,,  =  -'H-      (■  n't!  I  IS  MrH>ns.  .;  increases  as  th,    ..nfue 

ai'  a  AuA  the  he.ul  thminish. 

1  he  f.liowin;;  are  ^onu-  of  the  Min(htions  ^lH^h  t.  nd  to 
II.  'dily  the  vahie  of  <,  : 

(It  Tlie  contraction  is  /////>,/•/,•,/  .ind  will  |„  supprisseil 
ov.  r  tin  lower  e(l(,'e  of  a  sijuare  orifice  .it  tlit  bottom  of  a 
Vessel,  and  over  a  side  as  well  if  the  orifice  is  in  .i  corner.  In 
fai  f    t!ie  contraction  is  more  or  le-s  imi)crfcct  for  .uu  ..ti*",cc 


Li^ri/icii .\r  ()/-  (OXTK.-icT/nx. 


.^5 


within  tlirrc   .li.inuUis   !V,.ni  tin-    ^u\v  ,<\    hntton,  nf  the  vessi'l. 
Ihu^,   the    I  rns^-sei.-tii.ii    of  th(    \  fiia  mntr.u  t.i   is                    j 
iiUTcascil,    .111(1    r\])triirKnt    vho\\s   th.it    tile    ih^-                    j 
ihaiLjc  is  also  increased.  i . 

iJi   I     i'^  incre.ised  cr  (liniini'ihed  .m  (irch'riL;  as  j 

the    siirl.iee   siir|-,,undmL;    thi-cnirKc  is   eiin\T\.,r  ~l 

(.iiiua\e  t"  the  mil  linr  iifthe  \e.sel.  ^'i'-   23. 

(3)  The  e.iiitiaetinn  is  iiiiin  rfeet  and  ,  is  increa-ed.  it  ilie 
oi-ifiee  is  place.!  m  a  ciulne.l  part  .,|  the  vessel.  ..r  it  the  uater 
ai)pr..aches  the  orifice  throii-h  a  chaniiei.  ns  in  Im'.^v  j;,  the 
Vfineity  ..r  the  lluid  rdanieiits  Ixmi';  ther.hy  c.  .nsulvrahly 
iiu  reas(  d 

'4  It  the  inner  nh^v  of  an  orifice  is  r.  .iiii<!e<|,  as  -.jicu,,  t,\. 
1-iL,-.    J4    and    j;.  the   contraction    is    iii.,re    or   less    iniiierleet. 


f 


(J 


Ki 


1:     LM 


I  Hi.    25. 


Th.    value  of  ,     van,  ^    l,.„„     ^4   f.,r   a    sha,  p-.-.i-ed   orifice  to 
'.cry  nearly  i,>/,n  (,,r  a  perfectly  romulfd  orifice, 

;»  Tlie  contraction  is  „„,■,>,/•/,/,  when  a  I.order  or  rim  in 
Pl-""!  round  a  part  m  ,|,.  .,|^,,.  ,,,  „„  ,„,,-^.,.  ^,,,■,,,^■^„^ 
uiuaids  ,,r  outwards.      Aciordint,'  to  Hidone. 


"    I     ;    •  1 5-    y  I'T  rectanj,Milar  orifices. 


.UK! 


<j)riiiah.\r  o/  coxtr  icr/ox. 


ft  • 

■''-^'   +  ■  1-''^     y  I'T  lirmlar  nriiui.- 


V  hoin-  th.-   IcM-th   .,(■  ,1„.   ,,l:^r,.  ,,,•  ,,,,   „,j,-^,^.  ,^^.,,,.  ^^|,j  ,^  j,^^ 
bord,-r  rxtcn.K.  and/,  tlir  |aTinu-tcr  ,  ,t  ilu-  ,„ilK,'. 

i>u    If  th,-    .i.i.'>   .,r  tlu    oritKoare   nn\r,l    .,,   as   t..t,„Mi    ,i 
bo!I-:n..uih    ■.!    tlu-   ]>n,|.,  ,rti,,ns   >h,,un    hs-    Im'l;.   j'.,  and    >,„rr- 


Fit;.   2'i. 

-spondin-    .ipproxiniatcly  f.    the    diap,-    ..t'thr    vrna    ...nttacta 
til.-  whole   nf  th.-   entrant, on    uill    tak,-    pla.ru, thni   tin-    ludl- 
in.Hith.  and         i,    unity  if  .h,-  a>,a    ..f  tlu-  nritkc  i.  tak,-n  t,.  he 
tho  an-a  ..t  tlu-  Muallcr  c-id 

I-or   M.d,    an    .,nC,.,    WVi.had.    ,i;ivc-,  llu-   Inllowni.^MahK-  ,,f 
values  ot  , .  : 


Hea.l    ivrr  Orifice  in  Feet. 
I  I  .4S. 

5  5-77 
>^7-9V 


-959 

'  '7  .-^ 


■|  hr  dimciisir.n,  .,|  th,  i,t  at  the  mn- 
!..i.t,-d  scition  ,,r  at  .my  .  ahor  jx-int,  ni.iy 
l)c  dircctlv  !n,-isurc.i  hy  nicms  ,,1"  sct- 
-.  reus  of  fuu-  pit.  h.  airant,re(l  as  in  I-'io. 
27.  1  he  -screws  arc  adjust,-, 1  «,<>  as  t-i 
t'.nch  th,-  siirfaci-  of  tlie  jet.  .m,!  th,  ,!,s- 
t.itu,-  hi-tucn  the  scrc\v-p,<infs  i,  (l„-n 
^ncasni  ,-d 


COHhhU.IFNI    Oh    (:OSrK.lf:TlON. 


7,7 


he 


MeasiircnR-iits  cf  \civ  pre.-it  accunicv  cm  he  marlc  witii  tile  ut- 
ini-asiir.-r,  Fii;.  2S,  (ksigacd  and  ((iMslrurtcd  la  liic  McCi.ll  laboratories 
wliuh  may   be  described  as  follows— One  end  >)i  a  horizontal  2-in.  bar 


V-i       h 


f 


Mi 


Fig.  aS. 

is  attached  to  the  front  <  f  ilie  lank  Um^.  ^S)  and  tlie  other  is  siq.iK.rled 
on  a  frame  l)oIted  to  the  sides  of  the  flume.  A  split  sle.?ve  sbdes  alonj; 
this  bar  an<l  may  be  c!ami)e<l  by  a  tiyhteninp  screw  in  anv  desired  |.osi- 
tion.  Upon  the  cross-h.ad  there  is  another  split  boss,  or  sleeve,  through 
which  a  second  bar  p.i8sos  at  right  angles  to  the  first,  ,ind  carries  a  siin 
ilarcrus<-hea<l  to  th.it  on  the  2-in.  bar.  so  thai  provision  is  made  for  a 
tonuh  adjustment  in  a  vertical  plane.  llirouKh  the  latter  cross-he.id 
j<.issesa  smaller  bar.  and  aloiiL'  tins  bar  slide*  a  third  adjiistahlp  cro^s. 
head,  or  calijKT-hoJder.  bv  which  the  caliper  can  be  swung  rnimd  and 
receive  its  final  adjustment.  Kor  ihe  measurenipnts  a  i:-in.  Rrown  \ 
^harpe  vernier  tal-per  is  used.  .\  capstan  head  ro,|  „  ,  h.mped  to  ea.  b 
I.-VJ  and  ran  be  swivelled  lliroi.gli  .,nv  angle.  .Ste.l  nee<lle-p..intcrs  .„e 
Mise.ie.l  m  the  heads,  .iiid  ,„..•  c.mped  111  such  position  as  inav  he  re- 
',niu(l.      Ill  making  a  nuaiurenient  the  steel  |>oint»  are  first   made  lo 


3'S 


coF.iiicii-:\r  o!  i>is(  hakcr. 


tuuch  and  t  ,.■  rorros,,o,Kli„^r  n-aciiMK.  ...U.-n.  The  points  arc  then  sen- 
arated  by  ..h.t.n,  ,h.  cahpor-hc.uls  apart,  and  tlu-'whol-  apparu  „s  s 
moved  uno  po„tu.„.  The  p,„„ts  arc  f.nally  ad,usted  s,.  a,  f  ,.u<h  the 
surfaces  of  the  jet  at  opp.„„e  p,„„ts,  and  roadin,.  a,.-  auain  r,^.'" 
iTonithe  t«„setsof  readings  the  transverse  dunension  nl  the  iet  Can  he 
at  once  determined,  to  ,he  or,e-thonsa„d.n  of  an  ,nch.   and  at  anv  point 

^^etween   -2   ms.  and   i   .„.    f ,   ,1„.    ,„ne,   M„larc  of   the  on.ke-plate. 

Hi^.hty  in   the  ap,,arau,,  ,-,  however.  n,.,M  essential. 

(./)    C.H-mcuut    <"    /V.v, //„/.;.,, —  If    O   i.    tlu-    cii^cliar-c    in 
cubic  feet  per  scconci  acn,..  tlu-  o.ntraetc-d  -.c.tion,  then" 

uherc   ,•    .    ,   ...    ,\    tlu-    CMHluic-nt    of  il|.J,.u-c   ..lui    i,  tn   he 
determined  1)\   cxpcriineiit. 


lunfl 


Fig.  j9. 


_  Intheoxpe,nn,-Mtsn,ad,-,„  ,1„  M,,,,!!  lah.  „a,orv. .  he  water  ,.n  le,v- 
I"8  the  ordtee,  passes  e„her  to  waste,  or  t.>  the  measuring  tank  throngh 
a    bifurcated    KaKan,/e.|.iro„    tubing.    ,upporte.i    in    a    p.vote.l    fratne 

unt.l  n  Mca.lv  he.Ml  h..s  been  obtained,  and  the  f.anie  is  thcr,  rapully 
«w..n«  throii.d.  a  snull  angle  bv  means  of  a  lever,  when  the  water  .as 
Ih.ough   the  other  branch   to  the  tank,      .As  soon   as  the  tank   ,s  siitFi- 


I 


COHhhlCIEM    OF  DISCH.-tRGF.. 


39 


he 


cicntly  full,  the  franio  is  swiin^'  bai  k  .,nd  tnc  water  ai,'airi  runs  to  waste. 
At  first  the  water  (li>cliars;eci  from  tne  tank  was  rejjlac  ed  li\  water  arl- 
nmteii  into  the  toj)  of  tne  tank  tl.roii-li  a  h^^e  ui  ininalin;;  in  a  ro-,e 
siibiiier^ed  just  below  the  surface.  .\itUou.s;U  tlie  uunust  can-  had  been 
taken  in  the  des.u;n  of  this  rose  to  ledace  the  eddy  niolioii  at  efflux  to  a 
iiuiiununi.  there  was  always  an  apfireriahle  disturbance.  The  li.iSi:  was 
liierelore  e.xtended  until  the  lose  lested  on  tne  Dottoni  (^1  tlie  lank,  ,S 
feet  below  tlie  ■)iirice.  with  this  ai  ran^eiiieiit  a  series  ol  onlic  e-flow 
e.\perinieiits  were  maiie,  the  tune  m  eaen  ^a-e  liem^-  t he  ine,;ti  (,i  ih,.t 
^jiven  by  two  stop-watc  he->  and  \\\<:  values  ol  the  >  oeliicieiils  of  dia- 
cliar^'e  are  ^iveii  in  Talties  .\  and   IS. 

'r.\i;ii;   A. 

1  Kl  Wi.l   l..\U    DRIIICIC    Ol-   .I),-,    sij.    IN.    ,\Ri..\    A.NIi    K  IM  A  1  .V  1  N(; 
OKIKICKS   OF   .im;j.-,   so    iw   .\l<l,.\ 


_:i 


in 
Feci. 


■4 
II. 
IS 
21) 


Circular. 


Ki|iMl.ilcral  Tri.,  i  Rcclannlc  with     Rci  langlf  wnli 

. ingle  with       !     Si|uare  with         Vcrucal  Sides      Vcrlu  iil  Siil.-s 
|Horuant.il  Hasi-    Vi.nu.il  bi.les.  i  K«|ual  in  Hour    Equ.il  t<j  .Sut.cn 
uppermos:.  ITimt-silie  Wi.lili  Timtsthe  U  lUii 


T 

'173 
.')l3 

.(iIO 

007 

.(,..4 

.(103 
.(102 
.li02 
.(.01 
.001 


S 
.620 
.'1I3 
.  f>i)5 
■  hoi 
.(101 
.  fillO 
.598 
.=.,3 
.  =.).S 
•-<»7 
•597 


T 
.f.57 
.646 
.62S 
.623 
.621 
.618 
.bi7 
.(117 
.(ill) 
.615 
615 


s 

.63: 

.623 
.61(1 

.f,I3 
.610 
.608 
.f>07 
.607 
.606 
•  (105 
.605 


T 

643 
.63, 
.  (120 

■'15 

.612 
.611 
.6jo 
.609 
.607 
.(107 


62; 

62i 
.f.l? 
.6i2 
.  ()OI) 

.608 
.607 
()(>f) 
.6o() 

.605 
.004 


1 

.(l(:J 
.641 
.'.31 

.()27 

.'124 

.(.21 
J>2\ 
.620 
.(.I,) 

f )  I  s 

.(.IS 


S 

(.40 
.  ft2t) 
.620 

616 
.613 
.(,13 
.(III 
.610 
.609 

.008 


r 

.688 

■655 
.642 
.634 
.631 
.629 
.62b 
.(123 
.tl22 
.622 
.021 


s 

.671 
.657 
.643 

.63f> 
.632 
.629 
.627 
.625 
.625 
.(123 
622 


The  presence  of  the  hose  in  the  tank  w.is  not  sitisfartory.  as  it  nei  es. 

sarily  interfered  wilii  the  stieain-Mne  in m    .md  theiei.ire  allected  to  a 

Rieater  or  less  extent  tne  vaiiies  ol  the  c.elli.  lent  of  di.scli.ii^'i\  The 
hose  was  riiscaideij  and  the  water  i>  now  adnutteij  int..  .1  Much  i  h.iin- 
lier  cxtcndini;  riijht  .Kioss  the  bottoni  of  the  tank  and  c  ,  .111,1111111^  (ler- 
foratlons  on  the  lowet  Mirf.ee  throujjii  wh.rh  the  w.iter  Hows  to  the 
holtom  and  IS  there  detiected  U[.u..ids.  Twelve  inches  above  th,.  bottom 
'he  water  is  made  to  pass  thn.ii-h  ,1  b.ilfle-idale  perfor.ited  w.th  Mn. 
holes,  an<l  (1  inches  ln-hei  there  is  .1  sec.iid  h.,llie.i,l,,ie  ,ilso  |.erl.ir,ited 
"ith  Jin.  Iioles.  In  order  to  cijua.i/.e  .i>  much  ,is  |  .s-iblellie  l|..w  from 
all  points,  the  pitch  of  the  holes  ill  tlie  iip,,et  pkn,  »,,s  determined  by 
tiie  projections  on  a  horizontal  piaiie  .n  e,|u.il  distances  on  .1  sphere  „f 
loll.  diar.  with  its  centre  at  the  cento     .f  the  orilice  of  disch.it^c. 

There  are  two  outlet  pipes  lor  f.ist  and  slow  discharge,  .md  there  .ire 
1U-'  inflow  pipes,  the  one  3  ins.  ,ind  the  other  U  ins.  in  diamctei.  K.icJi 
f  these  pipes  I-,  controlh'd  li\  .1  .-t.  .p-\  aUp. 


40 


i:OEhhlCll-\T  OF  D/SCH.-lh'Gf-:. 


I.w.ia:  n. 

JjKIl  K   l>    nl     .1:1;    S.I.     IX.     ,\k1;A. 


In 
Heel 


Circular, 


F.quiiateral 

TriaiiKii- 
with  II'  ti- 
zontal  Side 
Ippcrmost. 


S(juare 

with 

Vernta 

Sides. 


j  RfCane  [Rectaiie.lRcctanc- 

Reclangle        wiih      ,      vviiti  wuh 

Square     wuh    Verli-   Venicai  1  Vcrntal    Vertu  , 
with     I    ,..,  Sides        Sides         Sid.s         Side, 
DiaKoiiali     Kqiial  to    ,  Equai  to-Equal   to, Equa.    lo 
\erl,ca.;ho.u     nnesj      6ne         S^xie.  m     One  six- 
|tllc  Width,     quarler       ' 
Width. 


2 

4 
h 
8 


T  six         s         T 

■  624  .6i,S    627  .6271.623 

.>tll>  .611^.620  .(121  .613 

.6ti)  .6o7|.6i5  .1,1;  .(,u& 

.607  .605;. 013  .613  .(104    .(,n 

.(>0(>  .(j<)4|.()12  (,I2   .f,i,3     .(|J2 

10    .606  .604    61 1  .(ill  .(,,,2 

12       .tK>^  .603    .611  (ill    .().,[ 

to4  .603  .()io  .6|().(ji<] 

606  .602.610  .610  .(K*) 

605  6(12  .610  .6ro    6<«)   .(, 

(mm  .601    .(riU)  (iIK,   .(,(M)     .(,1 


'4 
16 

iS 
JO 


S 
62.S 
6ji 
617 


Mn 
(ilo 
610 
6m 


S  !     T  S 

.6j3  .f,35  .(,40 

.Illy  .62f)  .633' 

.614  j  6i()  .62<|! 

.612  .(iK'i  .625! 

•  612  .614  .(125! 

•  '11  .612  .624! 

■  ''II  .611  (122I 
.'«)<)  (ill  .622! 

■  Ii'»)  .610  .620! 
.601)  .(j(i()  .(120! 

'loq  .(m>2  .620 


limes 
Width. 


S 
.641 

.632 
.62') 
.1.27 
.625 
.623 
.622 
.621 
.621 
.620 
.6211 


S 
.6?s 

.646 

•637 
.634 
.631 
.630 
.627 

624 
.624 
.623 

622 


teenili 
Width. 


S 

.(>sn 
.1.41, 

■(>37 

•6.-;  3 

.6u 

.62., 
.(,2'i 

625 
.624 
.62  1 

622 


.<  N.H.-  In  Tables  .\  ..„d  H.  T  in.lic.UeS  ,1  thickness  of  ,,latc  „f  .,(,.i„      , 

*  anil  .S  imlicaus  th.ii  the  orifice  is  sharp.cHijjc.l. 

Th.'  time  is  also  nu-asiirc!  cicctiicallv.      In  tlie  f,,rward  and  rctue, 
.nnyc-mon.s   the  Icv.r.  co„trn„in«  th,-  aii^.ilar  inovtnunl  of  tlic  ualv  ,1. 
./.  .i-.,„n  tubiMK.  "Kik.-s  aiul  l.r.Mks  an  c-lt-,tric  contact,  so  that  tht  inter 
^al  of  time  occupieil  Oy  an  e.x|.enment  is  registered  on  a  chronoRiai.h 
W  ih  this  new  arranKement,  tile  following  values   for  the  Coelfi   lent 

•■f  disch.ige  h.ive  been  deduced  I' ,1  sliarp-ed.ed  orifices,  the  area  ,n  each 
..-.l-iiiK  ptarticallythe  same.  vi/..    .„jt.y,   ..,.  „is..  and  equivalent  ,. 
the  .iie.i  of  a  circle  of  }  in.  .iianutei    — 


Iriati- 


•635'> 
.628'. 
.^>J2'^ 
.62(12 
6 1, Si) 
6lSi 
•fit77 
.61-6 
.6171 
■6l6i 
.(ilfii> 


S(| 

Jarc. 

Kei- 

*"*, 

liiar 

Keel. 

Head 

Katiu  III 

S'l 

es  4  :  I . 

Ratio  .>(  S  d>  1  16     . 

in 

CinuUr. 

.6|((() 
.6131 

Sidei 

Verllial. 

.62'i; 
.62()» 

OiaUonalH 
\"erli(..it. 

•  '1270 
.6277 

I.nns  Sid 

Vctii.-.r 

■"4ly 
■'■335 

Keet. 

'ti(f  Side 

"'/.■rv.i' 

l.nng  Sdr 

Vrrti.  a  . 

.(11,33 
.(,503 

.641.) 
.63l,.s 

6.U-' 
.^323 

t,une  Side 
Horizontal 

1 

■  6430 

.6644 
(into 

4 

() 

s 

ID 

.(loSl 
■60-3 
.6050 

.().ir'i 

.6|(,2 

•<'"37 
.(,127 
.MI6 

'■■77 
.6i;(> 
.ti3S 
.6.32 

.62S1 

.6255 

6234 

.6224 

.621)3 
.(<266 
.6252 
.6240 

.  6.|  I  e. 

.6372 
.634(1 

6.i2  7 
.6314 

63(4 
.62()8 

1  1 

.(i"4'l 
.6038 

MO() 
.6104 

.6123 
.6118 

.6217 
.621)7 

.62.30 
.6222 

•631  I 
.61(14 

1)032 

.()0((() 

.6.13 

6201 

.621'; 

■llTlOl 

.'K)3I 

.  t>oql> 

61 10 

.6200 

.6212 

•  (i2i)() 
.6291 

•  6293 

.62S? 

ao  I 

.  6U2i; 

.f>oci4 

.biuS 

1 

.6198 

.6210 

(:oi:mai:M  uh  disch.ikcj:. 


41 


At  liii'-t  twii  -it--  iif  nuM^urcnu  iits  wurt:  m.tilr  fni-  i  ,n  li 
licail.  and  tiu-  mean  w.i-  .Klopttd  ,is  cdrrcLt,  if  th-  roult-  d  d 
n<it  ililler  h>   more  tlian   ;  in    10,000. 

XiniRi'ius  c.\|)ii  iiiniu-  witli  a  i-in.  -hai-[)-e(i_L;cil  "vituc 
.qivc  .')  as  an  avcraL;c  \aliic  nf  tju:  li  k  rtRicnt  c  if  di-cliar-c  |nr 
luaiU  \ar_\in;.;  fmrn    I   to  jo  It. 

1  lie  jet  sprini^'s  Je.ir  I'mni  tiu-  ontice  in  all  case-;  repre- 
sented in  tile  alxive  table--,  .md  the  lollnuin-  inferences  Mia\- 
he  drawn  tmm  .m  in-peeti<in  df  the  same:  — 

(II  [he  coeffKieiit  of  dischar^'e  diininishes  as  thr  JiLad 
increases,  hnt  at  a  diininishiiiL;  rate. 

(2  1  'liie  cnefficients  U  r  ihe  tliick-jilate  orifices  are  in  ,dl 
cases  ;;reater  th.iii  the  c  nrre -j)!  indni^;  oeffu  iint-  for  shar|)- 
ed,t,'e<l  orifices,  excepting  in  the  ci-e  of  the  lon.^est  rectan^nilar 
orifice.  L'luler  a  head  of  i  ft.  the  coefficient  of  discharge  for 
this  orifice  still  exceeds  that  of  the  ^anie  orihce  with  a  -harp 
ed,f.^e,  while  for  he. ids  exceechn-  1  It.  the  coefficient  -eeiiis  to  he 
a  iittle  less,  hut  is  ])racticall_\-  the  same.  It  iine  he  noted  that 
the  thickness  of  the  jilati'  i-  j.  ;('i  times  tlie  wuhh  of  the  orifici-. 
and  the  coiitr.action  fir  the  ihuk-pi.iie  orhke  is  t  oii^eiiueiitlv  ■ 
iiK  ri  ased. 

1,^;  The  ccielfkiint  fir  nctaiiLjuiar  orifices  seems  to  be 
pr.icticaily  the  -anie  wiuiher  tile  longest  side  is  \-.rtical  or 
hori/oiital. 

i.|  ihe  coefficient  increases  with  tiu  ;u-<  a  of  the  orifice, 
exieptin^L;-  when  tin-  luad  is  \(.r\-  small.  'ihe  coeffu  i<i;t  for 
orihces  of  sniall  are.i  thin  r.ipidh    increases. 

(;i  With  rii  tan|_,nil.ir  oritlces.  the  coeffu  ient  iiu  re.ises  as 
the  width  of  the  orifice  diminishes,  i.e.,  ;is  tlu'  oritu  e  hecouus 
more  el(i!1j,Mte(l. 

1  he  two  hist  results  are  in  accordance  w  itli  sinul.u  results 
deiliucd  li_\-  Weishach,    Ihiff,    ,ind  o|h(  is. 

Note. —Tlie  manruT  in  which  the  he. id  of  u.uc  r  in  ihit.mk  is  dc  lined  i> 
both  simpU-  anfl  eflcciivc.  A  ^l.iss  K'ti'K^'.  "f  's  "i.  di.ir..  is  hxf<i  to  tlie 
tank  hy  iron  brackets  and  extends  from  the  top  to  the  bottom.     f)ti  one 


i 


4* 


liX.tMI'U.s. 


siilc  (.f  the  gauge  llicro  is  a  brass  scale  gradiLitcd  Imma  zero  |ioiiUiM  tho 
suiie  liorizoDtal  plane  a.-;  the  centre  uf  the  milice  ol  discliar^r.  A  ear- 
ner, witli  a  horizontal  wire  jiassiiig  in  fr.;nt  of  tiie  ga-ge.  slides  up  and 
down,  and  .,ny  rrijiiired  liead  i,  ohtained  by  briniiini;  the  necessary  scale 
graduation,  the  surface  of  the  uat<T  m  the  gauge,  the  wire  .md  its 
reflection  in  a  inirrrir  .,t  tlic  ha(  k  .,|  the  gaut;e.  into  tiie  s.mie  liori- 
z  au.il  [lianr,  'I'liere  is  a  siconil  indicator  on  llie  opposite  side  of  the 
tank,  coii:,isting  of  a  fio.it  .iit.ii  hi;d  to  ,iii  oidinary  water  proof  silk  li^liiHL;- 
cord  passing  over  a  huge  1  ght  Inctionic-s  piiliey  and  tlien  vi'rth.dly 
downwards  in  front  of  the  tanK,  The  cord  is  kept  taut  by  a  weight  at 
liie  bottom.  ,ind  c.irnes  a  friction-tight  pointer  which  can  be  easily  and 
lapidlv  adjusted  to  indicate  any  --eqiiired  in, irk  on  a  brass  plate  ll.\'ed  in 
a  convenient  position  on  the  tank  f.ice.  so  th.il  the  opei.itor  working  the 
valves  h.i-  it  constanti\  iindrr  ohserwition.  .\s  soon  as  the  he.ul  of 
water  in  the  t.ink  has  been  il.  terinined  by  nicins  of  the  gl.iss  g,-ui;'e,  tlie 
jiointer  i-  ni..ved  into  ]i(.sitioii  ..pp,,siie  the  m.irk.  ,1111!  is  kept  there 
throughout  the  experiment.  Tins  obviates  tiie  necessity  of  constantly 
watching  the  level  of  the  water  111  the  gauge,  which,  on  account  of  the 
height  of  the  tank,  is  often  very  inconvenient  ,ind  troublesome.  (  )ic.,- 
sionallv,  howe\er,  it  is  advisable  to  check  the  jio.ition  of  the  p. miter  by 
observing  the  w.iier-level  in  tiiegaugc,  .is  the  cord  indicator  is  e.\tieinely 
sensitive,  .ind  the  cord  itself  necessarilv  v,iries  slightly  in  !eii>.;ili,  •-, ,  tint 
small  errors  might  otheiuise  be  iiitrodm  rd. 

The  head  (jf  water  is  brought  to  .mv  reipnrcd  levil  by  means  of  a 
three-way  vahe  through  which  the  w.itcr  (.111  either  be  admitted  or 
allowed  to  escape.  The  valve  is  provided  with  ,1  loni;  vertic,il  spin- 
dle, upon  which  handles  arc  arrangecl  ,it  diflerent  points  111  sue  li  m.in- 
iii  I  ill, It  I'lie  1,111  l)c  easily  reached  .md  oper.iti  d  loan  .my  positiiai 
in  the  he!L;ht  of  the  tank.  Close  to  :he  1  ord  indicator  and  witliin  tlie 
reach  o!  the  ■  ■pci.itor  tlieie  is  a  mimI;  i-m.  pipi^  w  ith  \aKi\  wliudi  is  useful 
(or  a  line  .id  iiistinent  when  the  mlhiw  is  onlv  sliL;htlv  111  e.X( ess  of  the 
discliarge, 

l-.\.  I.  .\  vessel.  6  ft.  in  di.ir.,  is  full  of  water  ,ind  makes  tix)  levois. 
per  mm.  Find  the  velocity  of  etfliix  through  an  i;rili(c  1  It.  below  the 
surface  of  the  water  at  the  centre,  assuming  the  1  ieth(  ient  ol  \elocitv  to 
be  u>iil\ . 

Tlie  Ime.ir  velocitv  of  the  vessel's  periphery 


=  3»  =  3- 2 =  ".  i)ei  see. 

00  7 


Hence  the  velocity  of  elllux 


=/(T)' 


+  I. 32-2 


/■  \.-IMPl.tS. 


43 


./  48.400  ^,  , 

—  y  ^  I  j,S  =  33.4  ft.  per  sec. 

I^\.  2.  The  area  ul'  ,111  uriliLi;  in  a  tliin  |.lau-  was  36.3  cni.',  the  dis- 
>■  har<;o  uniier  i  hrad  of  3.31/)  m.  was  found  to  bi'  .01S25  in.'  iicrscr.,  and 
tlic  velocity  of  flow  at  tlir  conli:u:te(l  section,  as  deteriiiii)ed  by  nieas- 
urernents  of  the  position  of  thi'  axis  of  ilie  j(  t,  u.is  -.98  ni.  per  second. 
1  mil  llie  coelticients  i;f  velocity,  liischarge,  contraction,  and  resistance, 
taking^'  —  9.81  III. 


Therefore, 
and 

Tlierefore 
and 


7.98  =--  <„  4  2   X  9.81    X   3.396, 
Q  =  c.l  s  Iff!,. 


3<J-  3. 

(lOO)' 


.01825    ==  ,■    X      ~_^l,^  \   2    X    9.81    X    3.396, 


=  f<is<j. 


.   =  /   =    '"51 


.63J 


I 


c,  ■— 


.046, 


Kx.  3.  Thr  iei  from  an  orifice  of  ,008  s(|.  ft.  .irea,  under  a  liead  of  16 
ft.,  issues  liori/oiit. illy  .did  falls  1  !t.  vertically  in  a  horizontal  range  of 
~,G8  It.      I-";iu]  tl'.e  coetricurit  ol  veiocitv. 

•  -/'«)- 

f.     =    ,     =  .'^216, 

4  X   I   X  i6 


md 


'-,.  -  .'/.. 


l.X.  4.  If  .Cizj  is  the  I  I.I  ihiient  of  dischai^'c  in  the  prereflmt;  exain- 
ple.  tind  tlu'  di~i  h.iis^e  in  j.;aili>ns  per  minute.  The  oritii  e  Is  rectaii^iil.ir 
-md  is.2  ft.  wide  l)\  .04  feet  deeji  Find  the  iil-rhart,'e  when  the  contrac- 
tion is  suppressed  over  the  lower  <di,'e  bv  means  ..f  ,1  proji-ctint;  run. 

(J,  in  rub.  ft.  per  see,  =  .fijj    a    .008  \  2  .  32  ,  16  =  .16, 

and  thrref'  n- 

tiie  disrhame  m  }i.illon^  prr  minute  --  Uo    •   .16   x  6i 

:-  (10. 


^^  hen  the  Contraction  is  supjiri'ssi-d  nver  the  lower  edge, 


.9778 


the  coelt.  of  contr.ii  tioii  =  .621  1    -»-  .|C2 — 1  =  — — - 

V  2(.2   +   .04)/  3 


44 

'I'liiTcfore 


.Vf/,\7:A'-.V   I\(:h. 


Ilic  locll.  n(  (lis(  liargc  =  .</)  x     ■'^"'    =  .6338(XJ. 

3 
Ilriii  c  tin-  ilischaigf  111  cubic  fc-el  per  secoiui 

=  .633S96  y   ..,o,S    ^    r:2  .  32  .  16  =--  .162 
—  60,75s  ^'alliiii-.  ijcr  iiuiuitc. 

13.  Miner's  Inch.  //•.  („;,.  Sor.  C.  /■...  looov  -Tl,r 
niiiurs  mch  nf  water  i^  an  .ui)itrai\  lu.ululc  acioptcl  in  iiiininL; 
-listrict-.  tor  -cllin-  uater.  it  1-  \aiP,iisly  .Ictmol  as  hcinj  tin; 
;uiii.utit  ..f  water  cH-char-t.l  per  iimuitc  by  an  orifice  I  in. 
sqiiarr.  or  an  ciiiiv  a!/nt  fractioti  of  a  lar-er  orituc.  with  a 
head  of  troni  (,  to  .^  i„-,.,  tiu  tliicLnc-s  of  the  oritice  bein^' 
usualK   2  inches. 


y^^'  --'' 


CO 


-r^^^^LumLLmiiiuWj^^, 


Fi<;.  30. 

< 'ne  -reat  .HllKiiltv  i^  that  tliis  is  a  \;.riable  qiiantit\-  de- 
pcndniL;  np..n  the  .pe,  itie.',  Iica.l.  an.l  therefore  all  -~uch  iiukI- 
ules  shnuM  alM,  , I, -fine  the  \]n\\    in  cubic  feet   per  minute. 

'I'liere  an  ni,in\  praUi.  ,il  iliffuiilties  ni  the  n\  a_\-  of  ,leh\er- 
iii.l;  al)M,hitel>-  e\a.  >  .luantiti,-  of  w.iter,  b.  '  ihc  (lefuiition  ,,! 
the  nioihih-  or  nn:t  should  be  i  orrei  t  withiti  a  reasr.nablc  limit 
"''■''"•■■  •'  "  '-  •'  'l'-rnut,on  of  a  sin.Ljlc  miner's  inch  from  an 
'"■'li^''  '^t  '  -4-  "I  .  't  -lioiiid  -o  no  Inrther:  but  if  tile  mch  1^ 
de'inid  .IS    bein-    ,onie  fra.  tional    ;)art  o!   the  discliar-e-  iVorn  a 


Ml\f:R\s    ISCH. 


45 


lar-jer  nrifico,  it  ^lioul.!  he  liniit.-d  t.  the  oa|)aut>-  of  that  (irihcc. 
l-mtlicT.  a^  it  i-  a  tciin  ,>\  l.^al  -.i-iiitlc  atnui  onl^-,  the  dis- 
Jiai--c  ^hou!,!  i)c  Muc!)  in  ciil.ic  tcct  \h-v  minute,  eoir.enient 
(h-^chai-es  heiiiL;  lA  aiul  j  eii.  tt.  Ihellwu  uikIit  h  ,v,  hea.is  is 
irre-iiiar.  Head-,  ,.!  i  ft.  ,„■  ,n..ie  aiv  ii^t  Mutable,  because 
the  w.tter  i.  (K-h\eie.!  frnni  e'.itJie-.  (,r  tliinies  in  uhicli  the 
iK]ith  is  ne\ei-  L^re.it. 

Ihe  (|ue^Ci.in  tlm-  leM.he.  itself  into  ,i  choice  of  a  st.iiul- 
ard  module  or  unit  tVom  ,i  llou  un.lei  one  of  tuo  conditions, 
\  1/.  : 

■I  i  With  a  h.u  head  of  (  ^.  i„s.  above  the  centre  of  tlie 
ni-ihce  -ivin^  a  di-,dMr-e  of  i  ,■  eu^  ft.  per  minute,  \\  itii  the 
a(U,int.i-e  th.it  it  i-  ahead)-  pr.ictically  reco-tv/ed  ,1-,  the 
itimer\  m.h,  and  witii  t!ie  disadvantage  tli.U  the  tlou  i.  irreg- 
ular 

(Ji  Witii  a  head  of  1  I  A  ins.  ai)o\e  the  eriitre  of  the  orifice, 
and  a  thschaP^e  of  :.  ■  .,.  •  per  minute,  the  llo^^  iK.„ii.  mucii 
more  re-iihir.  but  the  (iuantil\-  disJuii-ed  i-,  not  reco-ni/ed  in 
pr.u  tice. 

I  h.'  tlnw  under  the  fust  conditi..ii  i.  eho.en  as  bein-  tlie 
one  nou  m  u-e  in  l5riti.ii  ( -..lumhia.  and  tlie  tolhnvmo  spec^ca- 
tion  is  -iven  of  the  miner'^  inch,  mcludm-  di-^char-es   of  fr    -n 

I  to   ino  miller'^  inclie->  of  \\  cu.  tf.   per  minute: 

'I'he  u.iter  t.d.eii  into  ,1  ditch  or  .luice  sh.dl  be  measured 
at  the  <liiJi  or  sluice  he.i.l.  It  sh.dl  be  taken  'rom  the-  m.dn 
ditJi,  ilnme.  or  c.m.il.  throu-h  a  box  or  reservoir  ,uran-ed  at 
the  M.ie.  ,md  the  uater  sii.dl  have  no  ..pprcu-.ible  vel-^city  of 
appro.ich  The  orifice  sh.dl  be  (ixc.l  \ertk  dly  at  ri-ht  anj^des 
to  the  .leliverin-  vvater\va>  ,  and  Hie  e.l-es  and  cnicrs  shall 
be  s,iu,u.'  ,ind  sharp,  the  top.  bottom,  an. I  sides  of  the  orifice 
hem-  at  n,.;ht  an^^h's  to  the  prcssure-bo.ird  The  issuin^^  vein 
shall  b.-  fully  contracted,  ,ind  the  ,iisch.ir-e  shall  pass  freely 
mto  the  ,m  .  The  distance  between  the  sides  and  bottom  of  the 
orifice  and  the  sides  and  liottom  of  the  waterway  shall  be  at 
le.ist  three  ,31  times  the  least  dimcnsi,,n  ,,(  the  orifice.  The 
miner's   inch  of  water  shall  mean  ,',  of  the  imanfitv  wl.l.-l,  cl,-,ii 


'^I\/'K  '.s    /,\v,7/. 


IM.n.cd        llH-    n,a.r    .1,,.!!    haw    , ,.,,„,t   ,,.,,|    „,■  , ,    ,„^ 

f"V'"'   "■"'"■    '"""■    '"■""'■•    ■""'    ""■    ~-'    'l->"ar^cd 
<h,il.  he  .'stimatcl  at    I  A  c.   ft.   ,,cr  ,„in„t,. 


I  Jischarjics    iq.   t.,   aiul    in>  l.niin. 


li  cu.    ft.   uf  \\.,t> 


"11.5;    miihi'-;  iiulic-,  (jf 


'   P>>:')i'H,i,   -,l,al!h,a~  intlu-  t.,li,,u  in,.;  tal.lc  : 


i)micnM„ns..f  (infkc.n  IncliM.        I     u     j        ,      .. 

,     H«.id  m  InihM       Xumbrrof  Mincr'i 
ov.rtenlreul        huhes  of  ,,(  ub," 


I 


Width. 
6 

12 

iS 
24 

■i 

6 

12 

18 

«5J 


3 
3 

a 
3 
4 
4 
4 
4 
4 


6.25 
fi  25 
b.z; 
0.  25 
6.25 
6.25 
6.  25 
6.25 


II 

.9853 

24 

.2485 

.16 

3S51 

4;* 

O.^fjs 

15 

G.;-)? 

2.1 

I'Sf'O 

47 

2953 

» 

02q6 

I 

5495 

'      '"■'"""•■  "-■    "A  Sr.,  l,.,sn,a,l..  .,n   intrrrsti.ur  scr.Vs  ,,r 

M.tUT.  J,nl,,,n,h,    ll^.h-anh.    I  ..ib.,rat,  „  ^  .  .M,  ,  „„  ^„i^  .,-.ity 

""•    'I'- !'■":;,  s    ,,.,,n|..,i    w.rc    ma.lr    nn.l.r    l,.w    hra.is    of 

•'■'""   •■    '"    i-'n,..,  an,|  u,th  tu,.k,n.l.  .,f,,MfK..s    v,/   ■ 

i.Mamiani     1^ '- ,,- ,1„  ,1  ,  ,h  tan,a!ar  ,  ^nficcs  in  1„  a-.s  „  ,  „„ 
'  '"4  sq.  ins.  mi  ana 

(2)  Squarc-c(l-r.l  i.c  t,.,,;;,,!.,.  ,„,ru.      ,M  u,.,„|     .  ,„.,    ,1,,   ■. 
-^t..4  ins.  in  hci-l.t.  .,,,,1  ^1  t,.  J.,  n;.    „,  u„ltl, 
""•  '"'"inl,!  ,„|.,,,t,.,l  |,„   i|„    ,!isj,.„^.,.  ,,,., 

L^     :    K^''»-.C(///   -   //,»;(scr.\rt,>i,.  .<j;. 
ill  uluch  C  is  the-  cnrmnrnt  .  .f  ,l|s,  I,art,'c  ; 

Cisthc  «listlinrj,'e  in  ^  nl.i.    I,  rt  ,„,  von.!; 
'■'i-  till'  ^M.ltl)  of  the  orifice; 
^/,.nul  //,,,]„    h.-a.ls  ,„..,  „„    „,,.  ,„„li„.n,,„,  „,„„. 
orifice. 
No  corr  .  Jons  were  nia.l.    f.,r  chan^-es  i„  tcui,..  raturc. 


MISr.R  W   ISi.H. 


47 


1 


'Hie  shajK  ul  ilic  nritKc  Ii.i-  ;i  vi'r\-  '-(•nsihlc  effect  iipf)n  tlic 
<li-i  ll.iri;;.-.  (  irml.ir  nrificc-  -ivr  llir  li  ist  (iiscliar.L;i'.  tlic 
^'rcitc-t  (li-t  h.irj^cs  occur  \mi1i  rccl.iiiL;iil,ir  cirilici's,  wlnl,  tli,_- 
tliscliarj,'cs  uitli  siinarc  i.rificcs  arc  intcniic.liati-.  I  he  >  .i.trii  iciit 
III  (lisJiart,'c  (C'  1  (liiiiini-.lics  as  the  si/e  <iftlii-  nnfu  .•  i;;,  rca'-ts. 
the  s.ime  Icirm  ^A  nni'ice  IieitiL;  inaiiitanni!.  I'm  the  ^.\u\r 
r'ritur  (  '  (hinini-,hr->  ,is  tlie  hiail  increases.  In  i  (h  t  i!iL;nlar 
orifices  of  constant  ihpth  tin  i  oefficieiit  of  .h-i  har^c  increases 
uitli  the  \vi(hli  ll  llir  ui.lth  remains  'on-tant,  the  coefficient 
nKreases  as  th(    di  pth  (hnnnishes. 

TIk'sc  cxjii  inucnt-  iMustr.ite  a  i -ai  i^us  point,  n,nnel\  ,  that 
various  -mall  oritur-,  j  in-.,  thuk  nni  K  in  .i  j-ni.  plank  i,  run 
full  like  ,1  slii.rt  tuhe.  .iiiil  tli(-.i  onticc  thrrrl«.ie  (lischai-e 
more  >\.itM  Ihni  th(\  tli.  orrticail)-  sjinul.l  if  the  \eni  w  ere 
'"'"'■"'"'  llie    ^-in.   X   2-\u..     i-in.    v   j-in.,    an.l     j-ni.    v 

.'-Ml.  on,  .'s  run  full  nndt  i  ilu-c  i..ni|itiuns,  .is  ,iUm  doc-  tlie 
I-in.  X   i-in.  orilii  c 

I  he  I-in.  V  2-in.  orilue,  j  ins.  thiik,  i^  ju-t  on  the  tn,UL;in 
111  twei  II  ll.iw  \Mth  I  ..ntr.n  tMii  ,mil  full-lioie  \\:<\\  .  It  it  i>  ti\eil 
III  the  \iiti>.il  poHtinn.  w.th  the  !Mii;;est  diameter  \irtical,  the 
^'ii'    i"ntr,ut  ll    It    i,    ii\(d  m  tlie    hori/ontal  position,   with 

till.-  longest  di.iini  ti  I  h'ln/Miit.il,  It  \m11  ,iIso  i-ontract.  i)ut  If  it 
is  rulihid  uilh  till  t'mj;crs  on  tlu  id-,,  ll  \\\\\  11,11  tiill  JMr  ,i 
time  .ind  thru  ,  out r.ut  .tjj.n'n  It  k.  pt  t  iiinim;_;  lull  in  tin-  w  ,i\  . 
it  will  disi  h.ir;;i-   .ilioiit    I   cii.   It     of\\,itei   pi  i   nnniiti'  niMic  ih.,,, 

W  111  n    lull    1  r.ntl.ll  thU!    t.iki    .    pl.i,  r 

'Ihe  j-in  \  J. in  orifii  r  run  ii.irtK  lull,  th.it  i-  to  s,n  ,  the 
loutr  hill  ot  the  i.ni'ui,  v\  here  the  i--uin;^  vein  turves  ilown. 
runs  liil!.  u'nlr  the  upper  li.ilf  contracts.  This  hirj,'cly  in- 
ire.ises  1  h  th<'  <hsi  li,ir^'e  and  the  coenTicicnt  of  (liscli,iri,'e,  hut 
tlu-  fl(.>v  iKcomcs  irrcj,'ul,ir  .nul  it  is  therefore  pr;ictic.dl> 
impo,sibl(  to  tne.eaire  .i  inipl.  miner's  inch.  l"or  this  reason 
,'^,  of  t'a  llou  (i.Hii  th,  o-in  /  2-'u\  orifu  e  u.is  chosen  as 
tlie  siind.nd  |i.i  tin  unit  niiner's  m.h.  ,iiid  this  niincr's  incll 
actually  clisch.irt,'es   i,49J<j  en.   ft.   j)er  minut.  . 


48  /,vi7  A-.s7(j.v  ('/■  jHr  Jin. 

\  14.   Inversion  of  the  Jet.      1  lu    pluiKuncnon  of  t!ic  inver- 

sion   <<\    tlu     jet    u.is    first     nwtiocii    1>\-    Hiddiic,    .iii<l    has    lircn 
subsequriu!\-  iiuotiLjati-.l  !i 


once  ct,  l.cstjros.  Mauiiuis.  Lord 


K.i\lei''ii,  tlu    author,  aibl  others. 


ScLtiunal    Elevjitlon, 


Cross-srctiitp 


Kii;.  31. 


Ki'..  u 


\\  hrii  a  iit  issues  from  ai\  oiill,  .•  in  a  vi-rtieai  surt.uc,  the 
sutiotis  of  tlu'  iet  at  poults  aloiiv;  its  j)atll  assutnc  siiinuUr 
lorni  .  'h  pi  lulrnt  upon  the  nature  oi  the  oritiee 

\\  iih  a  siiuare  .M    i  e>.  tan,;u!ai    onfiee    thr    srv  tion    .f  the    jet 

;,    a    star    o|    {oni     sliects    at    U',\Ut 
ant4le-  to  tlie  si,i,s.   l-'i^^s.   31.  3.'. 

3  3 
"1^^  Wiih   I  tiian.'.ular  orifice  the 

\       scitioii    Is  ;i   sl.ir   oi    three    slu-ets 

^     at     riL'lit     aiH^lcs    to    tlie     sulcs, 


Fie..  33- 


Kiii.  u 


'•'h;    u 


In  ^,'cneral.  with  a  pol\);onal  oiifue  ot  >;  side..  tl)e  section  i» 

a  st.tr  of  '/  sheets  ,it  iii;ht  ani;hs  to  tlie   .uli  , 


l\yf:KSl')\    ()/     TH/   JUT. 


49 


1  lu";t.-  ict<  friiii!  noti-i  in.  iil.ir 'irilKTs  h.uc  icntr.ii  inrc'-.  and 
the  ■^lin.ts  ,il  tl;c  ii.1,l;l--  ,irr  tliakcnccl  mit  mtw  hcuK,  l-'i^'s.  ^^ 
ami  ^^,  wlmli  arc  a])])i(  .\miatcl\  (■llii)tnal  in  sraiuii  with 
n'.ajnr  diaiintL-r^  i1miiI(1c  tlu  miiun'  iliaiiictcr--.  M.'.ny  (.-xact 
lllL-.i^iiri-nu-nt-^  of  tln-c  jrt->  lia\c  hoii  iiiaiK  .mi\  arc  ])artiallv 
(icscrihcil  III  a  pajHr  1)>  laiincr  aiui  .sti  it  k  lan.l  in  the  1  r.ms. 
C  an.    kiiy.   Smc,   \nl.    |\  .   src,    v 

With  a  -iinu  na  iilar  <a  ilicc  tlic  scctimi  li  is  a  iimrc  •  ir  lc<s 
--ciiiii,  IK  iilai  li'iuii'.l.ii ',  anil  a  -in;;lc  ^licct  at  riL'.ht  aivjlc^  tu  the 
uiaiiulcr 

I'lic  conimi.n  i  \  iilaiiatii  m  .,t  thi,  ;ii;cni  aiunMii  i-  that  tlic 
tluiil  |iaiticlcs  issuiiiL;  ahm;.;  li.H.iciit  paialH'h'.-  -trcani-hncs 
mil)m;4c  upon  c;uli  nthir,  aii.l  lis  th<ii  mutual  rcactHin^  cause 
tile    jet   t<i   s|)rca(l    '.lit    .111(1  a-vuir.c    ~L>.tii.nal    turiiis   Hcjjcnding 

U])i  111   the    aialH    111    till    ui  iliCf. 

I  iiu-  thi    lluiil   iMitklcs  issuing;  linri/.mtall>   ..mi  IncK    at  /.', 
■Aith  a    \iii"  it\     »  J.;.//.',  dcM.ri!)c 
a     ]).ii  all'  ila     /•/'  1  lie      [larlii  le 

issuing  at  I  w  itii  a  \  elm  it\' 
f2.(.'. /('  (icscnt.i  a  paialmla  tl> 
I'f  le--  1.  Ill  \Mtiii  e  than  /'/'  I'lic 
paitulc-  laiiiiii!  |)as  .  'iniuitani - 
iiusly  thiMUL;h  the  pmnt  /'  ami 
must  neies.ariK  pics-,  upmi  i\u  ii 
I  ither  I  111  >.     are    tllcreli  ire    i  >  'lil- 

pelleil  til  iiiiive  nut  v)l  then  iiatuial 
paths  anil  the  jet  spreads  into 
sheets 

A    tlienrv    whi:!)    s»-enis     ninre  '''"■    '''!• 

luliy  tei  acnumt  fm  the  uhnli  i  il  the  laits  is  tli.it  the  |k-i  iihar 
ihanpes  in  t.irm  aie  realK  due  tn  smlaie  tensmn  arid  tn  the 
diflereiu  es  lntXMin  the  atinnsphet  u  pressure  and  tlu  iiitcinal 
pressure  (it  the  jet. 

In    the    case,    fi-i     e\ain|)k/,    •!     s    jet    iluwins;     thnai^Ii    an 
elliptual  orifice  with  the  major  axis  vertical,  the   stream-lines 


50 


n.MF.    ()/    nu.lSi,   .4   LOCK. 


in  tlu-  VLin  an-  c-.nvci-cnl    uul  mutn.ill>  n-.ict  upon  cjuli  otlu-r. 
causing  tl.L-  jet  V,  i.mtract  vcrticall>-  aii.l  ,1.  :i-atr  h.  iri/Miuallr 


at    I  rate  ''ladiiall 


\   iiKi-oa-m.^  t.>  a  luaMiiumi,  w  Ik.-h  the  .Citicii 


H  a  ciiLk-  m  toiiii. 

At  till,  .ta-c  th.-  rate.  ..I  .•l.,n;;aliMn  an.I  contractinn  are 
llic  sanu-.  Tlio  cIoiiLjatiuii  and  c.ntractioii  still  rontinuc,  i;..i 
at  a  .iiniini-,],,,,;^  rate  initil  tli.-  ni.  .v  .in.'i.t  :-,  -t,,,,,,,.,!  In  the 
ctl.Tt  mC  varlao-  t. •„.:,,„.  u  lu„  tli,Mai..„  i>  a-ain  rlliptual, 
witl)  tlu-  i,ia|..r  a\i.  liMn/n„tal  and  tlu-  minor  a\i^  wrtKal. 
Tlu-  lu'u  major  aiul  inmMr  axe.  then  a-ain  l.o-in  n-p,c  tively 
to  eontract  and  to  elongate,  the  sccti..,,  ,,f  ,he  |c-t  passin- 
thr.)ui,'h  the  eircul.ir  inrm  to  it.  initial  ellipiual  torm, 

riiis  process  i,  repeated  over  tlu  ulu,le  l,n;;lh  of  the 
inil.roken  jet,  and,  ,n  taU,  in  thi.  ]  rtuu,  of  the  |et  i!u-  Mnlare 
tensi.ni  produce,  an  eCtect  .iniiL.r  to  thit  ^huh  u,„ild  !„■  p,,,- 
"iltued  ifthe  jet   were  .urrounded  h\    an  ela.ti;    en\(],,pe. 

l!  the  orilu,-  IS  small  .md  the  head  not  lar-e.  tlu-  |.-1.  ,„, 
leavin-  the  contracted  secti..n  at  the  onhce.  spread,  out  ml,, 
sheet-,  and  tlu-n  diiumishe.  t..  a  >ontra,  t,,i  ..-uio,,  .mnlar  to 
the  fir.t.  after  uhij,  it  a-ain  .pn-ad.  out  into  ,lu-.-|.,  I. Lectin- 
tlu-  an-les  betne.n  the  In  .1  ..-t  ot  ,lu-et..  and  a^ain  diinn.i.he'^ 
'"  •'  '"i't','^''-'l  .eaion.  Ihi.  action  i,  repeat.-d  .o  |,,n-;a. 
the  jet  remain,  unl.rok,-,,  A  .  oniparativelv  teu  expenuunts 
made  m  tlu-  !al,o,ator>   induale  that   it  tlu-  lu-a,!  //  i.    not  lar-e, 

tlu    :.;i:;-/i-»x//i     <  \  li     •    ,  . 

15.  Emptying  and  Filling  a  Caiial  Lock.  \Vh<-n  the 
}i.-.id  \arie..  a.  in  tillm-  or  empt>  in^  -  ,,  „  r.oir  .,r  a  hnk.  ni 
»illii,;4  a  ves.el  by  means  of  an  orifice  uiuhi  «ater.  ..r  in 
<mpt\in^;  water  out  ..("a  vessel  thr..u;.;h  ..  pout,  loiru  ,-|lis 
tht-ore-ii  is  still  employed 

II  tlu-  loik  or  vessel  i^  t..  he  tilled,  l-i-.  36.  h:t  .\  ..j.  ft. 
be  the  area  of  the  water-surface  when  it  i.s  x  ft.  below  the  sur- 
lace  of  the  outside  \\  atet 


F.X.^MPI.F.S. 


51 


If  tin-  lock  or  vessel  is  to  \n-  emptied,  Vv^.  37,  then  ,V  sq. 
ft.  i>  the  are.i  of  the  w.iter-siirl.iee  when  it  is  .r  ft.  abo\e  tlie 
orilice. 

In  eaeli  case  x  ft.  is  tiie  effective  liea<!  over  the  orifice,  .md 
is  tile  head  under  which  th.e  (low  takes  place. 


I  -I 


Frc.  36. 


Fi...  37. 


Ill    the   tiini'  ,//   the  uater-Mirfice  in  tiie  h.ck  or   vessel  will 
rise  or  fdl  by  an  amount  ,/i.       Then 

-   -^  .'/i  I'l.intit)-  which  iia.  entered  the  lock 

-/  liiin;.;  the  ,irea  of  the  "rifke. 
I  leiiie 

t=     /■       ^""^       , 
.  '  ,    cA  t  2gX 


an  eijuation  ^ivin-  tiie  time  ..i  lillm  .  or  emptyin-  tlu-  lock 
between  the  level  r  and  //.  Ihe  v. due  of  ,  |,,|  M.hmer^eil 
orifices  s.-en)->  to  W-  somewh.it  less  tii.u,  when  tlu  t1..u  -Kcurs 
frocl\-.  but  it  is  uMial  t..  tal  ••  .0  or  .f.j;  a.  a  mean  value. 

Ex.  I.  A  ii.ir.ilxiloitl.il  vensrl  with  ,1  laiiis-rpctui  1  of  1  It  and  5  ft.  in 
lieiKht.  is  immcrsoH  in  water  to  a  drpili  0(4  feel.  H.mv  Iomj;  will  n  tak.- 
to  till  Ihr  vessel  to  the  l.-vel  ..f  tl..-  oiif*ide  nurfaro  through  an  (.riti(<- 
I  inch  III  ili.ir.  al  the  vrrte-x  »     '.  Tike  .   —  5  1 

l^'t  left,  le  the  dial  ..I  llir  fi.-r  surl.K,-  wlu-n  il  is  .  ft  atuive  rlie 
orifice.     The-' 


52 


i:.\.-i.u/'/j:s. 


Aisu,  Utile  WiitLM  riso  ,/i   ft.  iii  ,//  si'^-s.. 

.T)'(/.r  ~  aim  Hint  crUi'iing  vossol  in  ,//■  sees. 

—  quantity  fl'iwm'4  tiiri)Ui;li   critin-   under   a  liead  of 
4  —  f  ft.  in  ,//  sees. 


5         I       I 

,,  T  .      .  ^  2    3:(4   -    I  ).  ,//, 

h        4     144 


and  llieiefiire 


y  .  d.x  =  .r,/.r  —   -'-(1  _  .1  )*,//, 
57t> 


<// 


576 


1 


5      (4—11 


5      I 


4(4-1 


3?^'    (  4  -  (4  -    r>  ) 
.    * 


./i- 


14 


\ 


,/i 


Tnte^'raliiiL:    between    the    liniits    .t   =  o     ,ind     .1         4  (t.,     tlie    refiuiied 
time  111  sees. 


57"   \  J 

=     .      -,4.. .4' 

=  i:2,S.S. 


,i    ' 


Ex.  2.  Tlic  li.ui/..nt,il  sn  tion  •,f  ,1  |.„  l^-i  li.iinhrr  is  .-ippri  xicalelv  a 
rertan^ilf  and  its  Icuf^tli  is  3(10  feci.  I'lie  m.Ic  a.iI!-  Ii.ivi'  .1  h.iiter  ( .f  1  in 
12.  .inii  till-  widtli  111  the  free  .-.iirf,!.  r  u  tieri  the  Ink  is  lull  ,,f  w.iter  is 
4^  feit.  IIk.v  liHi^'  \i.  ;!l  It  i.tkr  iM  iiiijiiv  tile  l.ii  k  thriiii^h  two  sluiees  111 
tlie  ^;.ile^,  im.  Ii  S  r  |i\  _■  |,  il,,.  sl.m  ,■  hoi  izunt.ii  1  entre-liiir  tieinj,'  13  ft. 
Iieliiw  the  fri-e  siirf.ic,.  m  the  loi  k  .mil  4  It.  Iielow  that  ^l  the  1  anal  on  tlie 
fiown  vtrcini  suit  "- 

Let  the  level  ,,f  the  walir  in  the  Lm  k  fall  ,1  ft.  in  /  seeoiids. 

The  area    jI  ihe  vvaUi -sui  f.n  e  i-  ilieu 


=  360(45  -    I). 


If  the  le\i'|  muv  siiks  ,m   Ii    m  U/  ^er«,, 

f  ^\ 

360J45  —  ^ji/.r  =  aii.i.uni   ol   water   whu  h   has  tliiwed  out  ihrouiih  tlic 

sluices 


=  2.^.2.8.  f'2  .  32  .  .»  . ,// 


GENER/ti    EQLIATIONS. 


5J 


Tlicrefure 


,//=9f^.,^-*_.    ..y, 


IntCKratinp;  helwccii  the  limit     i        o  and  .i        9ft.. the  rcquirC'l  lime 

in  sees.  =  —[90.9* 9 

=  600J. 


16.  General  Equations.  —  Hcrn<  nilli'--  tlunrrni  iii,-i>-  be 
casil}'  (Icdnccil  li  iiii  tlie  l;i  :kt.i1  fijii.itions  of  Htiid  motion,  ;is 
tollows :  — 

I.ct  />  l)i;  the  prr-^^iirc  .uid  ,>  tlie  diii-^itv  at  .u\y  point  whose 
co-ordinates  par.iliel  to  the  axes  are   v.    v,  c. 

Let  i< ,  ."■,  re  l)e  the  veh>eities  of  ilow  ;it  the  same  ]ioint 
parallel  to  the  a\e^.  and  let  X.  Y,  Z  he  the  acecleratini; 
forces  riieii  tin -e  equations  result  from  the  principle  o\  the 
f<[ualit)'  of  pressure  ii'  all  directions,  \i/.  ; 


I   '/-'  c/ik)  (in  (in  (111  du 

i.  ./.  '^      -     dt       "^     '    dt    -  "  dA      ^    ''dj      -  ''  ^-     ^" 


i>dy   ~  dt    ~ 


,>  d: 


dv         (h-  ,h  dv 

dt    -"dx-'dy  -'"'V--'     ^-^^ 

1  '^t         .,       "'•'''^        ^,      '''"''  '^•'"         '''•''  dxv 

=  -^  -     dt    =''^-  dt~   "dx  -  ''dy  -  "■  -dz     ^^> 


if  the  motion  !-,  steady,  d  that   the  velocity  at  any  point  is 

,,,//,  ,h'        (/:.■ 

.1  liMutioii  III  the  po^itiiiii  onh  .  then     ,-   =  O  --      ,    =  .md 

dt  dt         dt 

ii,  J',  -<■  nia\-  !)(■  i-xpressed   as   the  dilferential  coefficients  of  a 

iiinction  /■.       riuis, 


dp 

dy 


d: 


54 


CF.hlRAL    t-A^CATlONS. 


and  therefore 

liu   _    ,rF   _  Jv 
dy   ~  tfyiix   ~  t/x' 

du  _   d'b'  _  dw 
da  ~  dzdx  "  ilx' 

dj'   _  d'F        dw 
dz  ~  dzdy  ~  dy' 

Hence  (.'(lu.itioiis  i,  j,  aiul  3  nia_\-  l)e  written 

I  d/>  du  ur  div 

,     --  -V  —  /^         '     V        -   U'     ,    ; 
p  'IX  ax  dx  dx 


\  dp  dn  dv 

\   —  li    .  :■ 

ay  dy 


jj  ay 

I  dp 
P  d~ 


Z-H 


dn  dv 

d:    ~'   ■'  d.: 


ay' 

du' 

d- ' 


(4.) 


(i> 


iCn 


AIultipl\inp;  eq.  (41   by  dx.  cq.  (5)   by  dy,  and   eq.  (6)  by 
</^,   .md  .uldui;;,   then 

'^Z  ,     ,  ,.,        I       . ,  ,  ;'/-'/     ,  <//<  du       \ 

-^  ^  Xdx  ^-  \  dy  -f  Zdz  ~u\       dx  -I-    ,  dv  4-    ,dz\ 


:(/u 

\d. 

Id: 


,      ,    dx  ■(-    ■ 

\dx         '    ,/i 


du    ,  du 

dv  ,  ,/.-  ,  \ 

dy    I      ,  ./r 


-/-      / 


wliicli  may  lie  written 
dp 


Intc<jrating,  and  as-^uininL;  llie    hud  tu  he  houin'^ciu-ous, 
^^        I  (.V,/..    (    )  ,/r   t-  Z,/c) 1— y-! f-  a  constant. 


LOSS   ISI   SHOCK. 


35 


Ihnci,  il    L^r.uity  is  the  uii'.y  t>)icf,  .iiul  il    / '  is  llif  ri.---uitailt 
vilocity  .it  the  point, 


and  the  la^t  fquatiwn  becnmcs 


u    p  .■  ■-)-  .V   —  I    ; 


P 
P 


y 


>ii-tant 


V 


=  —  .C~  —  ~,   +  '1  t'>^-^t.mt ; 


and  tlurcforp 


■  + 


a  constant. 


17.  Loss  of  Energy  in  Shock. — An  ahmpt  ch.iti^a-  ot'  sec- 
tion at  any  jmint  in  a  lciiL;tlt  ni  pipin-  (ic-tri>_\s  th:.-  paral'icli-^ni 
of  tlir  tliiul  lilanuiU-,  lirtak-  up  the  lliiid.  ,nul  t;iK-i\L;y  i-  dissi- 
])atcd  in  tile  [)rc.dia  timi  of  eddy  and  otlRr  nuitions.  'I'lie 
tncTi;}-  tluis  wa'-lL-d  is  tLinRd   ■•  iinrxy  Ifi!  in  ilioik." 

A- 


Fir.    38. 

In     A    -llnrt     lrn;^tll    I'T    plJiMlL;-,     I'lt;.     .^S.     W  lu-lc    the    scith'n 

sixldcnl)    cli.uigcs  from  ,/  /.'    tn   /./.  .  i>nsid(r  tin    iluid   mass 
bftwcrn    til''   two  tran--' ftM    -iitioiis  J/;,   wlnri    tlir   mo;ii,n  'if 


56 


LOSS    l\    SHOCK. 


the  tluiii  fil.imcnl>  !ia.^  l)rcn  uii.li^tnrhc.l  .iml  i^  in  jKuallcl  linos, 
atiil  ('/\  uluTc  tlic  paralKli^m  lia^  hci-ti  a,;,Min  ic-i'-tahlislicd. 

Ill  an  iiuictinitcl)-  -^lioit  interval  of  timu  /  let  the  nia-^  nio\tj 
forward  into  the  ])o-,ition  huiinded  h_v  the  plane  >ei.tiMn-,  ./'/•' 
ami  c" /)  . 

Let  (?|  ,  T'l  ,  /-I  he    the    seetional    area,   \el'ii,it\      4    ll<iu,   and 

iniMn  inten^it)-  ul' pre^-^ure  at  .  /  /.' . 
Let  (7, .  r-,.  /•,  he'  ^iniil.ir  -^ynih')!-,  tor  (  ' /' 
Let  J,.    ;■_,  hi-    tile    elevation    ahii\e    datum    >  ■!    the  (' .  (i.^  of 

tile  sectional  art-.i^  at  ./  /■'    aiiij  i" P  . 
Let  //  he-  the    \-ertieal    di^ime    hetween    the    C   (i.  -    oi    the 

an-a-.  /-./and  .  /  /.' 
Let  y  he  the   mean    mteii-itv  .it"  pie^^nre   <  .er  the  annular 

siniaee-  hetween   //and   ./   /.' . 
Tlie  n.-sultaiit   force  aetmL;    in  the  dneetion  .il   m    tion  u[)on 
tile  mass  ot  lluid  under  consideration 

—   u>niponent  of  uei-lit  of  mass  ui  tiiis  direelioii 
-[-  pressin'e  on  .  /  />' 

4-  pres^un-  on  annulai'  siirfict-  hetween  AA'aiid  J  />" 
—  iH-e--sure  on  ( "  /> 


=  :.vr,./:f^ 


.   J- 


a?snminL[  that  /'  r=  /<^  ,  or  that    the  ni<  an    mten-ity  of  pressure 
is  uneh,ui-ed  throuL^diout  the  wiiole  ot  tlie  ^ectiim  /•./•. 

riu-  iiiirm.d  reaetmn  cif  the-  pip(--->iirtace  hetueeii  /:/-'and 
(  /'  has  no  eoniponent  in  the  direction  of  motion,  and  tric- 
tional  re->istane(  s  are  ihsi  (.-Ljardc  d 

Hence  the  impulse  of  the  it--ultant  tnret- 

-  :.v;,(:r,        j_,  -    //  /    ,    „  ,i /-,        />.)/ 
=  clian^e  of  momentum  in  the  ^aiiie  diiection  of 
the  fluid  masses  (  7V>  (     and.//./.'./',  -ince 
the    monu-ntnm   of  tlu-   nia-s   hetwe-eii   ./  A' 
and  C  /'  remain-  i;iu  han-.-ed 


LOSS   IN  SHOCK. 
JO 


3l 


-■.( 


,;'^'V  'i^ 


=  -  ^.(Tv  -  r,r,V, 
A"-    -     - 

sini    ,  h\-  till.'  ronditiMi:  of  t untimiity, 

Ifu'idin^     thr()iiL;lu)Ut     !)>■    ;lu-    l.ictor    riv?.,^.    the    equation 
bccunics 


\shu!i  iii,i\    be  written  in  the  I'lnn 


j: 


1'  : 

A"" 


Xow  tlie  pipes   are   near'\-  alwa)'^   axial,  and  in   such   case 
//  :^  o.  so  tliat  the  list  eijiiati'iii  lieeonies 


':+;:: 


If  there  Iiad  tjeen  no  abrupt  change  of  section,  or  if  the 
cli.in,m  between  .■//.'  and  I'P  liad  bc':n  ^'rathial,  th(.'n  no 
internal  woik  wouKl  ha\e  Wv\  done  in  destrovini;  tlu'  p.irallei- 
i-ni  ot  ;he  thiid  t'llanients,  and  no  energy  waste'd.  Therefore, 
by  Hernouilli's  theorem,  the  relation 

'     '     re    •     2,L,-         -    '    Ci'    '    2^ 

woul  '  lia\  e  tlien  held  j^ood. 

I  V    —  V  ,  J '' 
I  hu--        '  "ft  -11)^    of  enerrx'  tier  pound  of  fluid  is   the 

2g 

If's.'.  IV  sliiuk  bet\\een     /  />'    .md  (  /*. 

l^xiierniieiit  justifies  the  ,i>siinipti<in  /'  =  /i,. 


5« 


BORIK-IS    MOLTHP/HCE. 


V.\.  At  a  (K.int  .7.  150  't.  al)"Vc  (lau.rn,  a  line  (if  inprnf;  --iiiiili'iii v 
<ti)iii)h's  Hi  -.t-i  tiiii:,il  ai  f.i.  Il  tliL'  vcluiiiy  ol  \]"W  n  tin  larf^rr  inpc  i> 
J>  ft.  pi-r  >ci  .,  .iiiil  1;  t  hi'  ]irc---uu'  .,t  .  /  is  1  j;  Wis  per  mj.  11;  .  linii  tlic  prts- 
surc  prr  S(|.  III.  at  A',   N  It.  atnA  e  (latum,   the  indlioii  lie  111 '.^  steady. 

Tile  veiociiy  (if  t'.ow  m  tiic  Miialler  pipe  is  evahril  Iv   !'■  It.  ]ier  se(  mid 
Tliercfuic  llie  loss  ui  lie.nl  m  ^lioi  k  al  liie  sudden  eliaiiKe  of  setliuii 


(16  -  ,S)-^ 


1  ft. 


Ilcnco,    il  /  Is  tin-  pres-.are  per  s(p  111.  ,,t  /i', 


8   + 


/■    ■    144 


or 


and 


t^'4 


+   I   =  I  50  + 


12;   X    144         K,' 


62i 


64- 


62- 


43^ 


f  =   i,S7i  Wi-.  per  s(j.  in. 


18.    Mouthpieces i,/i    /i,'n/,i's     Mcnthpiccc.  — This     is 

mcrcK-  a  short  pipe    jirdjcetinL;   inwanls,  as   in    l-'ig.   yj,  which 


Kk;.  v\. 

represents  a  jet  llouin"^  tlir(Mu.;li  ,.  re-entrant  nKnithpiccr  of  set - 
tional  area  .!,  t'lxctl  m  tlie  \ertuMl  side  of  a  vessel  of  constant 
horiz'int:!  section  anil  cfrntainiriL,'  water  kept  at  a  Kinst.iTU 
level.  1  he  mouthpiece  is  suftu  iently  loni;  to  allow  (if  the  jet 
sprinL;in;4  clear  froir,  tlie  erui  A /•  without  adherini;"  to  tiie  in-^ide 
surl'.Ke 


hOHIh-l'S    MOU'WriECE. 


59 


Tilt'  \i'ln(.it\'  III"  the  lluid  inolcculcs  alon^''  AC  and  /'A',  is 
sufdiuntlN  '^111, ill  to  he  ilisn'^^irilcil,  so  that  the  pre-, sure  over 
this  ]iiiitioii  o|  the  vessel  is  (listrihuti'd  in  accord. nice  with  the 
h>-drostatic  la\\-.  The  same  may  .dso  be  said  ot' the  pre-sure 
On  the  rcniaiiuicr  of  the  vessel's  surface. 

Ai^ain.  the  only  unbalanced  [iressure  is  that  on  the  surface 
//(■/  immetliately  opposite  the  inoutlii)iece,  aiul  the  resultani 
horizontal  force  in  the  directicjii  of  the  axis  of  the  mouthpiece 

=  (A.  -f  -^'^'A  —  /i„,-J   =  u'/iA, 

h  beint,'  the   de])th  of  the  axis   beiow  the  water-surface   and  /> 
the  intensit\'  of  the  atmospheric  pressure 

1  he  jet  con\er^^es  to  a  minimum,  or  contracted  section  ,J/.\', 
fjf  area  a. 

In  a  unit  of  time  let  the  fluid  mass  iietween  //)'aiui  J/.V 
take  up  the  position  bounded  b\-  A  11  m\k\.  MX  .        i'heii 

Xi.'hA  =  imi)ulse  of  force  in  direction  of  motion 

=  cluiiLje  of  momentum  in   same  direction  in  a    unit 

of  time 
=  difference  between  the  momenta  of  .1A\'.\'  .1/    and 

Ar^H  A  ,     since    the    momentum    ol     the    m,.--^ 

between  .7'/)'  and  .l/.\' remains  uncliaiiLjed 
=:  momentum   of  J/.\'.\' J/  ,  since  the  momentum  of 

A  i';/>  A  '  is  \  ertical 

7» '  7t ' 

r=       (/.'  .  r    ^=      ii:-, 

r  bein^^  the  mean  velocity  of  tlou  .icross  the  contracted  sectuin. 
I  leiu  e 


.-.■//.•I    .. 


''  .  -.i,''/'. 


and  therefore 


.}    =    2,1. 


\J.    V 


60 

or 


liOKUAS    MOl   IHI'IHCE. 


A 


cotltR'iciit  mC  I iiiiti ,k til >n. 


rili^  result  ll.ls  Ihi'II  \iry  i  l(i-,(:ly  \cnt"li';i  l)y  t,'\  ptTI  inctU , 
the  fiicDu  lent  Im'iii;,;  \wvu  tuiiml  to  he  .514'y  l)>'  |{iilil,l,  ^£;4~ 
1)\-   Hill. me,    .i!ul     5  1J4   liy  Welsh. kIi. 

H'TiLi's  niiKithpn-i  e  -iM-^  a  -.maher  (li->eli,ii,L;t'  than  a  sliafp- 
edijed  .inlne.  hiit  a  ili-ihaiL;e  ulireli  is  iiuu  li  iiMre  iiintnrin, 
a'l'l  heiu  e  It  1^  ;^eiierally  n^eii  ill  vessels  t'loni  \Uiuii  v\attr  is 
I'  1  he  ili^trihiit.d  h\    ii\e.i-^in  (■ 

N'iM..  — Let  I'il;  4(1  re])i'-ent  a  let  l1ii\wii;^'  tliroir.di  a 
re-entrant  nii  Mithpiei  e  of  ^el  IMiial  area  ./,  lixc.l  la  tlie  -.lopmi; 
sidi-  ot  a  re^er\-.'i|-  ■  .  int.uniii'4  water  k-pt  at  a  i  on^taiit  li\cl, 
and  sujjposf  tliai  the'  le-ri'.mi  is  dt'  -.m  h  -i/e  that  i,IIKI  may 
represent  a  eyhmlrual  lluid  mass,  eo.ixial  u  itli  tlie  muiitlipuee 
and  so  lari;>  th.it  the  \elnui\  ,it  its  surl.iee  is  sciisitilv  ml. 
Let  //',  //  he  \\w  dc  pth-  hel.iw  thi'  u  ater-surtai  e  ot  the  C.  G.S 
ot  tlic  areas  f///.ind  AV  ,  respi\  ti\el\-. 


i 

♦ 

-y.  - 1 

t' 

i 

K     1 


«•, 


KiO.  40. 


Then  tlif  nsiihant  force   aIon|,r  the  axis  of  the  mouthpiece 
■=  i>ressiire  on  (,H  —  pressure  on  ^A' ,md  on  /'/ 


RIN(]N<)/./l.h.  0« 

—  prc^-^urt'  on  /•. /• 

-|- C(iinp(iiu'iit    of    tlir    Wright    <<\'   the    tluiil 
ni.i^-  (,ll Kl 

—  (^^^  _|-  7,.//')  .irci  CI!  -    (  /'„  +  ..■//      .ire. I  (  'A' 4-  area  /'/.  i 

'■'  -  ''' 
—  />„  .  area  //■'-{-  Te  .  area  (,//.  dh  .         ...   ,  very  tu:arly 

1  lnu  e.   111  a  unit  cif  time. 

ii'///l  ^-  iinpiiNe  <>\  till--  t.inc 

=^  I  li.iiiL'e  ii'  nil  iiiietit  ,mi  in  direiti'Hi  nf  axis 


=  -    iir  .  ;•  =        tiv-  =         (I  .  2_i^^/i, 

(J  luiiiL;  tlie  ari  a  of  tin    ei>ii1raet(  d  >eetinn,   u  liijc  It  is  also  very 
ai)i)r(i.\iniatel_\   the  di  ptli  m|  it--  ('    (i.  hejow  the  \vat<T-siirfacc. 

Thus,    .IS  l)e|i  ire, 


the   I  Leffa  lent    >'\   I  (Miliar  tlMii    =: 


J 


{t>\  Kiii,i;-'i(>::li         I  In    i  in:^-ii.i//le      >ee    l-'ii;.    .i  i  i   is   ulnn 

Used  \\lth  .1  t'lrc-eilj^llle    let,   .illd 
^"^  I'OIlsist-.   ot    .1   re-etltr.int     pipe     I'l 

seetiiiii.ii  are. I  ,i^  li\< d  in  .i  jHpe 

/  />.,  ^  /^-  lit      ^ettiiiii.d      .uea     ii ,.  I  he 

/' /  •  <,'"  leii'Mli  ot  the  re-iiitiant   piutmn 

-.IK  h    that    till     \^   iti  I     .pi  111;;'- 
I  h  .11    hi  an    till     mill  i    end    .ind, 
\\  ithi  lilt       .iL;ain      \-  an  Inn;;       t  he 
Mill. ice      o(      the       iiiouthpieee, 
'■'"'•■•''■  eoiner^jc-     t"    .i    ninhmum     or 

COIltracte(1    se(  finn    >i|   .IMa   ,/   .it      I/.V  . 

C'on^idei  th<  Hinil  m.is,  lietwi'cn  1/ \' .iiid  .i  ti  .insverse  sec- 
tion /l/>'.  .Hid  III  .1  nit  o|  linii  lit  It  ino\(  into  tli,'  pii-.iti.iT) 
bountleil  by  tiie  |.l.m.--  ,1/    \     md     /  /■'  . 


62 


RINO-NO/./.l.E. 


Il  i--  assuincil  that  tlii-  niwiioii  i\  -,tc.i(K-  .uul  tli.it  tlu-rr  is 
till  illtc  rii.il  \\  .  >\\i  (liK-  t'>  the  pri  iiliu  tli  III  nf  ^'ciilir,  mi  ntluT 
nioti  iii>. 

l-L-t/,,,  .  In  till  iiifc'n-,ity  lit  the  atniD^plnTH.  jjh^-^uil'  ami 
tlu'  M'loiitv  at   .1/  \ 

I. ft  />!  .  r'l  Ih-  tlif  iiR-.m  intfn-'it\   I 't"  pir-Min    and  tli<-  \  rli  luty 

at  .//; 

I.ct     /'    111'     tlir     nu-aii    iiiti-ii--it\'    m|    tlu-    |ir(->iiir    .i\,M     \\^^^^ 

aninilar  sni  lat  c  /•  /• ,    ( ,  / / 
1,1't    ;,,,      I  lir    tln'    ili\  ath 'I1-.    aliii\c  datimi  .it    tli^'  (',   (i.^.  of 

tin-  •^ritmii^  .)/.\   a'lil  .  //,'. 

IllCM 

^  impiilM.-  ill  (lifciti.m  i)t' nidti.iii 

--   chanjjc  of  ini»niciitiim  m  •^,lllu•  iliriitin!!  in  a  unit  ut  tinir 

=  difrcrcncc  <it  tlu   nioPR-nta  nf  tin-  iIhh!  m.^^M'^  .I/.\'.\'  .)/    ,inii 


I*"  -     ' 

A'^^uniin^'  that  /'        />,  .  tlu-  I.i-it  (ini.iti't!!  Ik'v.>mu-> 

;*'/?.^,U,   —  ::„;    !    //,* /•,    -    A.'  =    *', '"'"'  —  'V  ,■')•    •      *»» 
H\    Hcrnouilli's  tin  iin  ni, 

s    A-   ^'     4-    '  ''    -  -    4-   ^"   4-    '- 

-I  -r  .^,   1   ,_^,  -  -..  T  .^,  -r  2jr  • 

.iHil  till  ri'I'.n- 

/,  -  A,   _  '■*_-J't* 

X«)W  ^,  —  r„  is  very  >-ir.all  and  nia\-  Ik   iiisrcj;arclfil  vMih'.ut 
sensible  trrur.  and  tluii,  t)\   ^■i\-^.   i  i     ,iii<l  (Ji, 

il;  -  'V  _  /*!  "  A.  _  •  "t''  -  Vi' 

2^  "•  A'  «, 


I-') 


C  Yl.lM>h-IC.4i    M<  )L  ■  I HPILlH. 


63 


\\v 


*  1 


(rt,'  -  «-):•,■• 


f/|  iV,''  —   it.,7\'  i'f'f>'  —    It'll, )i\'  il  <!,' 

■-1IU-1-  (/,.-|  =  <ir. 

Il   till'    -^ccti' iiial    ,irc,i  (/,<)'"  tlir   iiipi-'   i^  \ cry  I.UL;r   a■^  coni- 


I 


I), 111  ■!  'A  ith    ,1.   --I   tl;  il 


i.i\    Ik-  ili->rcL:.iriii'il  witliniit  ■ichmIiIc 


vrior,  tile 


.   ,1111!  tllrrcti  ire   tlic  i  ■  irtln.  uttt  1  it    1. 1  mtrai,  tmil 


<l  1 


,1--  111    I  >oi  1 1,1    .  ni'iu 


tlipici.  c. 


{<)    (  i/ii/t/r/i  ii/    Moulli  [^un  .  —  W'luri     w.itrr    i--.iifs    frnni     .1 

c\'liiiilric.i!    iiiMiitlipun  c   (scr  Vv^. 

42)   at   liM^t    t\'  .1   til    l'.\i>  ..111!    nllr- 

lialf  (ii.miitiT-  111  l'-iiL;th,  tlir  jet 
i»iii'-.  tiill  III  ii  1  ,  I  ii  \\  iilii  lilt  \  i  iit- 
1 1  ,K  1 11  111  J I  tin-  pi  lint  111  ili^i  li.u  l;c. 

JL 11/  1.1-  '111-    >r>tiiiii.i!    aic.i   111' 

'hi-  nil  iiitlipu  I  r.  -',  till  ill  ptli  ol 
il  <  a\i.  l)(|ii\\  till-  \\  all  I -siirtai'c, 
am!  (_'  till'  aiibiimt  lit  tin-  ilis- 
c'liarL;i'.  thru  ixpriimmt  --linws 
that 

C>=  .S.vJ  1  j.c//.      .      (I) 


/i 


m...i... 


Fi...  ».-. 


■|'!ii-  I  KiHkicnt  S.'  i-  the 
pii  iilin  t  I  il  I  111  1  1  n  Hu  Hills  I  il 
\'fiiH'it\'  am!  1  1  uiti  ai  tmn  .  Imt  tln' 
coefficient  nf  cnntracti'in  i-i  nnitv ,  .uiil  tlicicfoiv  the  iiKltuit  iit 
of  vtlni  it\  u  S;?  Now  tiu'  imai)  cuitVKii.Mit  of  vclocit)-  in  tiu- 
cast  111  a  -.iinpli-  sharp-i-d^ctl  orirtcf  i^  947.  .in.i  tin  difforiiue 
!ifi  \\  cell  1)4/  .iiiil  >ij  (.iiiniit  l>i  wliiillv  .ninmiicil  Im  l>\  iiic- 
tional  resistance-*,  Ijut  is  in  |)art  tlue  to  a  loss  of  lie. u!  In  t.ut, 
the  \vater.  .is   it    1  'e.irs  flic  inner  Cil^je  of  the  tHoiitli|iin.  c,  con- 


64 


CYl.lSnh'lC.-t!    M(  )l  THriFXE. 


VLTf^cs  to  ,1  minimum  cctiun  .l/.\'.  nf  ana  ,?,  and  then  swells 
nut  uiitil  at  .'/  .'^'    it  a;^M!n  fi!!--  t!i-,'  moutlijiiccc. 

KtUTf^f)'  is  wast-fd  in  i(i(l\-  motions  hctut'cn  .!/^  aiid  .!/  X  , 
wlu'rc  the   action    w  simil.ir  to  that  uliich  occur-  m    ,U)ii;))t 

ch.iiiLjc  of  section. 

Let  /,  r  he  tlie  intensity  of  the  pressure  and  the  mean 
velocit>'  lit  tlow  at  the  iioin!      t discharge. 

Let  /•,  ,  ,■■,  be  similar  syniiiols  |nr  tlie  .    ■■tractid  secti<'n  .I/.\'. 

Let/,,  be  tlie  iiitensit)    of  the  .itmospiienc  press    re. 

KeiiiemlH'rin!.'    that    —  ' is   tlie  lo--    of  he, id  ••due  to 

-V,'' 

shock    '  between  .l/.\'and  J/  .\'  ,  then,  b>  lie.  .iouiUis  theoreiii, 


2S 


^g 


Hence 


and 


A  -  /       ^'i' 


a' 


^S 


-    //. 


^2) 


(y) 


2jr  ( 


vhere  c    =:  eoeffiueiit  oi  ^  ontrai  ti'Hl 


'I  iiereli  ire 


'  1 


V 


2g>h    t 


1  + 


P,    -  P 
w 

u.  -  -) 


) 


(4) 


an  <ciii,itii>ii  t^Hiny  the  velotil}  ••!  iImu    it  tlic  point  ..f di-n  h,ir"e. 


CYUSnKlLAL   M(  )l  THru-ch:. 


65 


If  til.    (lisc!iarf,^c  is   int.i   llic-  atinnsplicij,  p,,  —  p  aiKi  r(|ii.i- 
tion  (4)  bccoiiios 

where 


C,-    2gh,       . 


(5) 


I  -^ 


xV. 


.      (6) 


If  c  —  .6j.  then  C  --  .85,  uliilc  cxpciiment  ^'iv<-'--  .Sj  as 
till-  value  of  C.  I'lvJ  Miiall  (iiffereiu-e  between  .H;  an  1  S2  is 
piohabK  chir  to  frutmnal  ri'^-i^taiiLe.  The  value  ,Sj  l"r  C,. 
makes  C   apiimxiniatelv'    '117. 

Aj^ain,  tlie  <liscbari;e  from  a  -,iniple  -.harp-etl^'cil  (iril-ae  ot 
same  sectional  ,iu  a  a^  tin-  Mouthpiece  is  dz.-l  j  2,;'/'.  '"'  niore 
than  24  per  cent  le-;-  tli.ui  tlic  ili->ihar^;e  troni  the  i  \  hmlncal 
UMuthpiece. 

1  lie  loss  (>t  hea.l  between  .I/A' aiul  M  .V 


1,r 


// 


'.-■; 


//  X.i276  z_  .487  X 


2A' 


,  ai.ptn\nnat<' ly. 


Thus  tl-e  effective  head  is  only  ^/r,  .nstead  of  // 

Hy  iq,  (31,  the  difference  between  the  piessure-head  .  at  .I/.V 
and  at  the  point  of  discharge 

■a-  2^' 


=  J//,  ver\'  n<'.ir!\'. 


66 


nii'l  KCI-ST  MOl  ThrirCF. 


-i. 


\ou  ifcino  i-iifi  1 'I'  .1  tuln;  1^  iiiscrti'd  in  the  niuutlipiofc  at 
tin-  iMiUr.utiMl  •-(•(.ti'.ii  i  l''i.L;  42)  ami  llie  utlicr  end  iinnicrscd 
in  a  \c^scl  'ir\\atci-,  the  water 
will  at  iMue  rise  to  a  heiL;ht  k^ 
in  the  tube,  ^Iihwiiil;  that  the 
j)re-.^iire  at  the  i.  ( intraeted  --ei. - 
tion  i>  le■^-^  than  that  due  to  the 

atni' i-pileri'        !!)■  1.  all-till  inei>-        

iirenu-nt  it  1-.  f'Mind  that  Z/,  is 
\-ery  nearl\'  eijual  to  ii'i,  wIulIi 
verifies  tlie  theory. 

o/)    Dirt  rji^^i  lit  .I/( '//////"/< •(■<•. 
— Suppose  that   tor  the  c>  lin- 
(Iricai  mouthpiece  in    <  »  lliere  is  sul)Ntiiuted  ,1  di\er;;ent  nioutii- 
piece  of  the  exa.t  form  of  the  issuini;  jet,   I'i^.  .\\.       I'lien 

[\  1    The  moiith])iei.e  will  nm  full  bore. 

{i\  I'here  will  he  no  joss  of  Iieail  l)etweeti  the  iiiiiiiiiu  11. 
section  .l/.\'  ami  tlu  |)lane  of  discliarj;i'  .//>,  as  tlure  i-  11  '.s 
110  alirupt  vlian^e  of  section. 

Ileiui'  h\  Bernouilli's  tluoix-m,  and  retainiiiL;  tin  ^ame 
^\  nihois  as  in  (11. 


\'\'..  n. 


A._,    ,      A 


.'  + ; 


rn 


If  the  discliari;e  i-  into  the  atiiiospliere.  /'  —  /'„,  and  therefore 

;•■  -     2,Q-//\ .' 

or,    illtnxlui  Hi.,    a    voelVlcieilt    i',.(^  .<>X.    nearl\,   loi     a      in"oih 
well-formctl  mouthpiece^, 

and  the  (lisdiari'e  i-> 


/  t  -'cA. 


•4) 


DiyHRijHNT  MOU'IHI'lEcn.  6? 

I'Vom  tlie  last  equation  it  woultl  appear  as  if  the  tiiseharj^e 
wiiuld  increase  indetiiiitel}-  witli  ,/,  but  this  is  inanifestly  im- 
pii---.ilili,'. 

In  fac;,  by  ecj,  ii  i,  tlie  tl'iw  biiiii^  into  tin:  air,  and  taking 
'      -   I, 

^^^■^-  '■^(^•':-.) (5) 

={■:-"(■;; -0 (6) 

>iiKc  a:\  ■-—  Ar.       Hut /•,  (aiumt  tie  negative,  ami  theretorf 

'    ■"(•"-■). 

■>■.  that 

A  /p 

T  ^V^wh   •   ' ''^ 

pivcs  a  niaximum  linu'f  fi  m   the  rafin  'if./  tn  ,? 

p 

Nn'A  —  ^4  tt     \ri-\-  m.uU'.  anil  the   la^t  inuati'in  nia\'  bi- 

written 

A  34        h 

a        \  h ('^^ 

\\\  eii>.  (4)  an-l  (7), 

0=  C.^s^l  ■iv.yi   .    ^'),       ...      (9) 

\\liieli    is    alsn   tlu-    cypir-^^inn    Im-    thi     di^Lhar^'e  t!iroui;li    tin: 
miniinnm  sei'tion  a  nit"  a  '  a>  num. 

II.  henvevcr,  the  sectional  areas  dl"  the  niiMithpiece  at  th<- 
pi  Milt  of  (lisciiarKc  <iiid  at  the  tliro.it  are  in  tlu  i.itio  of,-/  im  ,/, 
as  j;ivei;  b\  rep  1 7 1,  it  is  founil  that  tin-  full-boie  (lou  will 
be  interrupted  either  by  the  discn^af^enient  of  air.  or  bv  any 
slight    disturbance,    us,    for    i  .\,inip!i',    a    sli^dit    blow    on    the 


68 


CONyi:RG{:XT   MOL'IHnFCn. 


iiicuitiinit'CL'.  and  Ikmicc.  i:i  pr.ictic',-.  it  is  r.-^ual  to  niako  the 
ratiu  ui  .1  U>  a  scusibl}-  loss  tlian  tliat  l,m\'-"  by  eti.  171 

((•)  Convergent  Mouthpieee.  —  With  a  ciiiivci'L,'ciit  inMithiJici.  c 
fl'"i^T    ^.^  I  two  piiiiit-^  art.'  to  he  noted; 

(I  I  'I'lien:  is  a  contraction  withiii  tlu-  .iiouthpiece,  followed 
by  a  swelhni,^  out  of  the  jet  until  it  again  fills  the  mouthpiece. 


r 


Fic.  44. 
Thus,   as   in   the  ra^e    of  cylindrical    nioutli]5leces,    tJiere   is    a 
"  loss  i)f  head  "  hetvMenthc   lonliactrd   section  andtlu■;^•mt 
of  (liscliarL;e,  and   also  a  consciiiu  in   ilinuniition   in  the  \clo,  iiy 
of  dist  har^e. 

(J,  There  is  a  secoiul  i  oiitraclion  outside  thr  ini  iithpirce 
due  to  the  C(>nvert;ence  of  the  iliiid  filaments.  1  he  mean 
velouty  oi  flow  !,■■  i  across  the  section  is 

C,',     hvWV^    the    coefficient     of   veloMtN-    and     /'    the    effeitive    head 

aho\<'  the  lentie  o!  the  section. 
Also,  the  area  of  this  seition 

—.  c^'  X   are, I  of  ni'iiithpiec  e  at  p..iiit  of  di>i.  harf;e 
r,'../. 
cj  beinjr  the   coefficient  of  contiaiiion        liem-    the  disci-   rge 
Q  is  yiven  by 

Q  =  e^'i,'A  i  zgli  -  ,  A  \.  2.;//, 

c'{—  r,,V,')  bein;^'  the  coeflu  lent  .il  disciwirge. 


Mi 


ES'l:RGY    AND   MOMtlNTLWJ    Oh  JET. 


69 


The  cncffiLicnts  <  and  r  dcpoiui  upon  tlic  anfjlc  of  i:<^\^- 
vcrsjcncr.  and  Castcl  found  that  a  i.-on%-crL;cncc  of  1  ^  24  ^r^ve 
a  m  xininni  disclr>.r,L;c  tliroui;!)  a  nimithpiccc  2  -G  diameters  in 
Icn^'th,  t!ic  sin.illcst  diameter  beini;  .();t)S5  foot. 

TAHLK    (.iVINC,    CASIKLS    R[:>rLTS. 


Angles  nf 
Convt-rger.ce. 


0  o 

1  36 

3  ro 

4  10 

5  26 

7  52 

8  ?S 
10  20 
12  4 


C  ' 

e 

.999 

I  .CXXJ 

1 .001 

1.002 

1.004 

.998 

•  992 

!    .987 

.9S0 

.830 

.866 
.894 
.910 
.920 
93' 
•  42 
.950 
.955 


Aneles  of 
Convergence. 


829 
866 
895 
912 

924 

929 

9.14 
938 
942 


"3° 

-4 

14 

25 

16 

36 

19 

28 

21 

0 

23 

0 

29 

5S 

40 

20 

48 

50 

.953 

•979 
.969 

•953 

•  945 

•  937 
.919 
.887 
..)6I 


.962  i  .946 
■966  '  .941 
•971    '       .91S 


.970 
.971 
•974 
•975 
.980 
.984 


.(124 
.>ii8 
.913 
.  S96 
.S69 
.847 


19.   Energy  and    Momentum  of    a    Jet. — [a)  Jit  from  a 

sliarp-t  J:^,-  ,>rifuc\ 

-J 
1  lie   energy  of  the  jet  —   riv?.-'         ft.-Ihs    per  seeoiid 


•ar' 


=      ^       ft.-l!)-^.  per  ••ieeond 


—  7<.'<i:li(,.'  ft.-Ili^.   ])er  -(H  onil 


pave 


I.,  p.. 


/(=  :■■//!  hcini;  the  iisdrostatic  prcssiite  (hic  to  the  head  //.  and 
tile  a\era;.,H-  \ahii    of,      heui"  .62. 


1  he  nionientuni  of  the  jet 


S 


--    2pilC,\ 

and  tins  i,  eipMl  to  tlie  pressure  in  pounds  produced  1)\    the  jet 
af;ainst  a  fixed  plane  jh  1  |i(  ndii  ul.ir  t..   it-  dueetion.      Nci^Kct- 


70 


FXAMrl.FS. 


iiv  ,■  -',  the  thrii-^t  is  (loiih'j-  tp-  'i_\-(in«statir  prc-^-^urc  dm-  t(>  t!ic 
head  //. 

(/'')   ./"'  f"'"'  •'  Cylinilrual  MouthpiCiC. 

Tlic  i'KT;4y  of  the  jet    -    7K\\r\  -  ft. -lbs.  per  second 
—   -  ft.   Ibv.  i)^r  ■  ei  I'lul 

■>ir 

^  i,;'u'A  ^2x/r  It  -Ihs    per  second 
-=^  "^ h.  I'.. 


the  averai^e  \alui-  of  i  .   beini;  .S2. 
The  nionieiituni  of  tlic  jet   — 


\'lv  .r  ^  u'A   '  —  2ZL'Ahc,}. 


Ex,  I.  Water  tt  '\\>  tln.ir,L;lia  !?i.r'i.i  inoinli|iiri  c  of  ,/  si],  ft.  '^I'C- 
tional  area  utuliT  .i  hra.l  .-f  /■  Irrt.  il  tlir  i>t  si>ru>;.;s  dear  ft.a.i  the  inner 
cdj;e,  liie  (iiscliar^je  is  29.2?  less  .,i!il  llu-  jil's  (  ner^;v  4i-4  •  K"''"<>'  '''•'" 
when  the  inouthpiere  runs  full. 

l.et  r   he  I'ne  nii'an  vc'.x  itv  d  0  'W  ;u  i"--  \\><-  '  oiuraitei!  sectuin  A/.V; 
u  be  the  mean  velo,  ity  of  tluw  at  the  mouth  tP  wlien  tlie  mouth- 
piece runs  full.      Then  7/  =  2U. 
Let  C^i,  /--'i  be  the  'liM  har-e  aiul   energy  of  tlie   jet   wh.ii    ir   -.prints 
clear ; 
Qt,  A"j  be  the  dischait;.'  ami  eiie   ^y  "f  tiic  jet  when  the   mnuili- 
piecc  runs  full.     Then 

.;  


in.i 


A,  = 


When   the  moutlipieci    run>  full,  the   loss  of 
heail  biiween  .I/.\'  anil  (  /' 

(T'  -   II i'  _  ir 


Fic.  45. 


Hence 


o  +  —  +  -    + 


•H,' 


IL 


fiX.lMI'IMS. 


7' 


u'  =1,'//  —  — . 

A  2 


Tlu'rcfure 
and 
Henre 
AUo, 


O,  =  .lit  =    — lV, 


/:, 


71/Q,  u'      It' A       v^ 
.707      in.l      -  -^-^         =  .292 


.414. 


£■.       4  t  2  A'l  —  A"j 

-7"    =     -  1.4  T  4     .iiul  ,. 

/',  4  ^■■' 

\:\.  z.  Dcli-rinmc  tin-  (li^cIlarJ4es  aiul  energies  of  a  jflumlcr  a  lie.id  of 
100  ft.,  issiiin},'  from  a  6-iii.  mouthpiece  which  is  (u)  cylniiri.  al.  ['",  di- 
vcruiMil  ihcll-moiuhl,  if)  convery;ciil,  the  an^lc  of  coiivorL-cncc  Ijeiiit; 
29»;S'. 

(a)     ?■  =  .S2  |/04  .  Idj  =  6;/j  ft.  ]>cr  sec. 

O  =  "-'  (  '^-V  X  65.6  =  I2»icii.  ft.  ]icrser.  ^  So' g  gais.  p.T  sec. 

7  4\i2y 

62i  X   12U  (65/')" 


Kiicri;v  = 


54.152,',  ft.-lbs.  =■-  <jmi  11.  1'. 


(fi\     ?■    -  .98  iV,4.  100  =  -.S.4  ft.  [icr  '■•ec. 

Q  —    ''    '  (     '    )   X  7^.4=^  15.4  cu.  ft.  per  sec.  =  96i  yals.  per  sec. 

KiRTL;y   -  -         -   ,,     '  ,         =  92.43S;,  ft.-l!)-.      -    l6.S,^„  ■!.  1'. 


n  - 


Knerj,'v' 


71.68  —   14.08  cii.  fl.  pcT  .-ec.  =.=  88  gal>.  per  sec. 

-064-.807  ft -U.S.  -   r;S.4;   1!    1>. 


(f)      XI  =  .S96  ♦  (14.  100         7I.<'i8  ft.  pel   s<'C.      (S.r  Ce-li  1'^    I'.ible.'i 

2;    '  /  ^' 
7    4VI-' 

6.;'   .<_14J'-   171  ^'•'^1' 
32 

I'.X.  V  There  is  a  36-fl.  head  "f  uatei  I'Vcr  tin-  2  r'.  lliroat  of  a  bell- 
mouili.  l-'upl  the  yieatest  (li:mn-!it  (.1  the  Tii.iiith  when  open  to  tfie 
atiuo-pheie  aiiil  miiiimi;  lull,  th'  hrii;ht  oi  the  water-ljarumeti  r  Ijciiig 
34  fe.  t. 

1,1  I  /■,  :■  tie  tlir  pros.  aii<l  vel.  .it  t  he  thioat  ; 
;/    hi-  the  vel.  at  tlie  iiioulli. 


Then 
Thcr'^fore 


f- 


r^'^^''^  V^  '^■ 


f'4 


or     7',  =  8  4/2  ft.  per  sec. 


MICROCOPY    RESOLUTION    TEST    CHAR' 
ANSI  ond  ISO  TEST  CHART  No    2 


iO    ifi^  i^ 


I.I 


11* 


20 


.8 


1.25 

1.4 

1.6 


Ji     /APPLIED  INAXGE     Inc 


1'^ 


72 


R.llil.4US(',    CHRRi.NT. 


Tin-  velocity  in  tlu-  ihr.Mt  is  yrc-atesl  wl,cn  tlu-  pressure.  />..  is  least. 
I.e.,  wlirii  /,  —  o.  .iiiii  llieii 

o  +    '      -:  ■>,(':     '>r     -■  -  48  ft.  per  sec. 
If  /;  iiis.  IS  th.-  .li.mieiei  of  tin-  inoutli.  (lie  .lis.  liarRC 

144     4  144       4 

eu.  II.  iier  set.. 


and 
or 


14 

1)1*  z^  12.726. 
I>  =  3. 56  ins 


20.  Radiatin>4  Current.  -As  an  applu  .iti-.n  -.f  Hernonilli's 
tluorrm,  ...uM.U,  Ihc  steady  plane  motion  ot  a  UnW  nl  watcT 
floxvin.^  r.uliall>-  bftucc-n  two  horizontal  i)lancs  ,r  It.  apart,  and 
svnnnctri.al  with  r.spcct  to  a  central  axis  (  Fi^;.  Af>) 

Lot  .•  It.  per  second  be  the  velocity  at  the  surface  of  a 
,  yl,nd.r  of  radius  r  ft.  described  al)out  the  same  axis.  Then 
the  volume  Q  crossing'  the  surface  per  second  is 


and  therefore 


(J 
rr  =  —  i*  constant, 

V7f 


since  0  is  constant. 

Tinis  .-  increases  as  ;  dunini^hes.  ami  becomes  infinitely 
.rrcat  at  the  axis;  but  it  is  evident  that  the  current  must  take 
a  i\e\\  course  at  si.iiie  fmite  distance  from  tiie  axis. 

If  /.  the  pressure  at  any  p"int  of  the  cylindrical  surface 
U    above  d.itinii    th.ii.  1»v  H.-rnouiili's  theorem. 


..J 


--I- 


1  const.int  =  A  =.l'-f-     ;. 


^ 


RADiATISG   LURRl'.NT. 


73 


dcnntiiiL;  the  ihnainu   lit-aJ  ■:  -T  y,  ^^Y  }'■      Hlt.cc 


.1  constant 


,uul  therefore 


r'(h      y        a  constant, 


T~n 


Kc.  47. 
is  ,ui  tiiiuiti.-M  L;iMn},'  the  free  siirfac.:s  of  tlic  pri'ssurc  K.himns 
tl-'iK    4"!        I'l'-''*'-'  •surfaces  arc  thus  tj.iiuratcil  by  the  rcvoUi- 
tion  of  wli.il  i-  c.illcil  Harlow-,  unvc. 


74 


i.-()Kii:x  Morios 


Tlu-  surfaces  <>f  cqu.il  pressure  are  also  -iveii  l)y  an  e.iuati.m 
lit  the  same  tunn 

21.  Vortex  Motion. ^A  vortex  is  a  mass  .>f  nitatiiii;  fluid, 
.1,1(1  the  s.irtex  is  leriiud  /r.r  when  the  iiuition  is  prnduce.l 
naturally   ami   uiukr   the   aetiuti    ..I    the    threes  .,t"  wei^'ht    and 

])ressure  onI_\'. 

Ill  the  radialini;  current  already  discussed,  assume  that  the 
directi..n  ..f  inotiMn  at  each  point  is  turned  thron^^h  a  ri-ht 
ant,'lc  so  that  the  mass  ,.t  u.Uerwill  now  r.vnUr  m  circular 
layers  al.out  the  eentral  .ixis.  ."so,  if  there  is  a  slow  radial 
movement,  so  that  fluid  particles  travel  from  one  circular 
stream-line  to  another,  it  is  assumed  that  these  particles  In  ely 
take  the  velocities  proper  to  the  stream-lines  wliu  h  llie\-  join. 
Such  .1  motion  is  termed  .i  free  ,/nii/iir  \orte\. 

■jhc  iiioli..n  beini;  stiady  and  hori/ontal,  the  eciuation 


-  4-       +    - 


—  =:  .1  constant 


// 


<0 


holds  -jock!  at  ever\    i.oint  of  a  liicular  str<.im  -.f  rulius  r. 
A^'ain, 

■:c  .  i/  (.:  -\-^)  ituremeiit  of  d>tiamic  ])ressure  Ixtueeii  two 
consecutive  ell  iiu  iitar>-  stream-lines 

=  tleviatinj,'  force 

=  ttntrifn^al  force  ot  an  l1>  iin  nt  li(tu<tii  tin 
two  stream-lines 

=  '  -  .  iff. 


Hut,  by  eq.  ( Ti, 


tfv 


Hence 


hREE  SriK.-ll.    yORTEX. 


75 


ami  tluiefore 


dv 


—  o. 


s.,  tlKit  vr  ^  a  constant,  an.l  V  v.irics  imcrsrly  a';  r,  as  in  ill- 
case  of  the  la.luuu...,  ainvnt.  ri.crefnrc-  the  curves  of  .(IluI 
pressure  uiU  al-^<.  be  the  <anie  as  in  a  radiatin-  current. 

/■n,   S/>tni/  I  ■<'//< ■-i-.—Supix.sc  that  the    motion   .>!   a   ma- 
nl  water  with  respect  to  an   axis  O  is  ..f  .uch  a   character  that 
at  anv  i-oint  J/,  the  c..mi.onem>  "\  the  velocity  in  the  direction 
of  ('.i/,  an.l    perpendicular   to  <'.!/.  ..re   each    inver<el\    pn.por- 
tional  to  the  distance  (U/  from  ' '.      The  me.tion  i^  thus  equiva- 
lent to  the   superposition  -.1   the   moti.ms  in  a  radiating'  current 
and  in  a  free   circular   vortex;    and    if  ^  i-   the   an-le   h.turen 
().!/  and   the   directi.m    of  the    ^tream-line   at    .1/.   .-  co>  H  and 
r  sin  «  are  each   nuer.ely  proportional  to  (  U/.  and  theret..re  ti 
,m,st    he    constant,       H-nce    the    stream-lnu-    imi-t    be    equi- 
angular s])irals,  an.l  th.'  motion  is  terme.l  a  free  spiral  \..rteN. 
"^  This  result  is  of  value   m  th  •   .liseussion    ..f  certain  turbines 
and  centr.fu-d  pumio.      A  sUo./v  free  surlaci-   in  the  case  ..fa 
free   spiral  vortex    is   imix.ssible.  as  tile  stream-lines  cross  the 
surfaces  of  e.pial  presure.  which  are  the  same  as  before. 

Also,    if  /.„,  r„.   .-„  an    tin-  pressur.-.  ratlins,  and  veloe't)'  at 
any  ..tlur  point  at  the  same  elevation       above  .latum,  then 


=  --+'r: 


and  tlu:  incrc.ise  of  pre-sure-head 

/■•,>r,o/  \',>rtix.—t\  forced  vortex  is  .me  in  which  tlu'  law 
of  m..ti..n  is  .lifferent  fn.m  that  in  a  free  vrtex.  1  he  simplest 
ami  most  useful  case  is  that  in  which  all  the  particles  have  aI^ 
e.pi.U   anK'iil.ir  velocity,  s..  that  the  water  will  revolve  b..dily. 


76 


FOKci-.n  .isn  coMriXsn  i-ortichs. 


the    VL!<ic!t>-  ;it    anv    jx.iiU   bcin^r   (!ir(,xtl\'    [impDrtimial    to   tlic 
lii^taiKc  truiu  the  .ix'is. 


I  p\        7-  ar 


\\n\. 


(.1  liciiii;   tlu-   const. mt   aiiL;ul  ir   \(lont>-of   the   rotating'   mass. 
'I'hiri'tore 


Integrating;, 


(         P\       fy* 
d\a  +  '  )  =      '- .  d>- 


J  -L  '    =  4-  a  (.(instant  --^         J-  a  constant. 

'    :.■         2,^'-     '  2.;'- 

lUiicc,  if/,,,  /•„,  ."■„  arc   the   pressure,   r.ulius,  and   vehuitN' 
ir  an>-  second  jioint  at  tlie  same  el'    ation  c  above  datum,  tlun 


P       P 
w 


•  r-  —  r,,-) 


(V-  —  V,,-). 


i/ 


2g  2g 

It"   tlie    second    point    is    on    tlic    axis    of    rcvohition,    then 
r,,     .  O,  and  the  l.ist  e<[n,ition  hcMiines 

P        P.  ^     '>'-j., 
W  2g     • 

1  luis   tlie   free  surface   of  tlie   i)ressure   columns   is  evident!)   .: 
p.ir.iiioloid   f)f  re\i)lution    with    it>   vertix 
.It  ( ',  as  in  Vv^.  4S. 

A  iKiiif.iiiiiif  \cirte.\  I'  produced  !)>■ 
the  combination  ot  a  centr.d  forced  vorte\ 
with  a  free  cireul.ir  voite.x,  the  tree  •sur- 
face beinj;  lormed  by  the  revolution  of  a 
Harlow  curve  and  .1  p.ir.ibola. 

l'"or  cxamj)le.  the  fan  of  a  centrifui^al 
])ump    draws    the    w.iter    into    .1    forced  Fn:-  18. 

\(>rtex  and  delivers  it  as  a   free   spiral  vortex  into  a  whiilpoo 
ch,mi»)er  'Chap.  VIII). 


V 


^^ 


r.XAMPl.l-.. 


In  tlii-  ch.inil.ir  there  i-^  tlui-,  .i  uam  "<"  pre.Mire-!ie;ul,  and 
the  water  is  therefore  enabled  to  rise  to  a  oirre^pondinij  addi- 
tional heiL^dit.  James  Thomson  adopted  the  theory  of  the 
compound  vortex  as  the  prnuiple  of  the  .laiou  of  his  v...tex 
turbine. 

Ex  A  rentrifntial  pump  ->f  ::  ft.  interior  and  4  ft.  exterior  diar., 
makes  3^'  revols.  per  iiurne.  The  water  gradually  fills  up  and  flows 
verv  slowly  througli  die  wheel  into  a  clianiber  of  con^parativciy  mucli 
larcer  diar.,  from"  which  it  pas«s  away  into  the  discharge-pipe.  The 
pressure  at  the  inlet  may  be  taken  to  be  <.ne  atmospliere,  or  21 16  lbs. 

per  sq.  foot. 

Hasing  the  flow  thtoui;h  t!ie  wheel  upon  the  livpothesis  tliat  the 
velocity  v  of  any  fluid  particle  is  directly  proportional  to  its  distance  r 


I'lo.  40- 

frnuitlie  axis  of  rotation,  the  law  eonnecting  the  pressure  /  and  the 
v.l.i,  ity  V  may  be  expressed  in  the  form  (Kx.  i.  p.  21J 

«'  -.V 

.\t  the  inlet/        3iiC>,  aii<l  lei  7'  -  7'i.     Then 
II  if)  ,    .    fi' 


so  til  at 
Hut 


:  n ')   ( 


««''  _  '="5/3'/^  i-,y_,  ,,,0,     i,nd     !-=':-]  =  r'. 


7H 


L.-IKdl-:    OA7/-7(;/f.S. 


Tlicrefore 

/  =  2ii6+  i;ii  !'•'—  I)  ~  906  +  i-ior', 

(iivinj;     r,     sikm  ossively, 

ihc  values  1.         I.-.  14.  '■(•■         i'^.       2  ft., 

the  corrcsponiliMfj  values 

,,f/,aro  :n6,    264,S.4,   3277.6,   4003.6,48264,    5746  lbs. 

Thus  the  curvo  .•//■'.  obtained  by  plottini;  these  values,  siujws  the 
variation  of  tile  pressure  inside  the  wheel. 

The  hvi>'>thcsls  of  the  How  ni  tiie  surrouudi.ig  chamber  is  that  tiie 
veliicilv  !■(  any  fluid  particle  is  tinvrse/y  proportional  to  its  distance  from 
the  axis  i)f  rotation  .  and  in  lliis  case  the  pressure  and  velocity  are  con- 
nected bv  I  lie  relation  (E.\.  1,  p.  21) 

At  the  wlierl  .mtl<-l.  i.e.,  where  '  -  2  ft..  /  ;=  5746  lbs.  per  .sq.  ft.,  and 
let  1>  —  7'j. 

5745    _.  ^.  _  TV 
«'  vC  ' 


Th. 


therelote 

Hut 

Ufi 

-AT 

therefore 

_  '-3/  1)"  -    ]  ^  ^s4o,     an.l  =      ; 


P 


574' 


19360 


4S4o(.  -■*,:^.o5S^-.'^ 


("living  >•.  successively,  the  values     2.     2.2.   2.4.     ::.''.     2-8.    and    3  ft., 
the  cf>rrespondinn  values 

of /are  574^'.   '"'SS''-   7^:25,    77^3.    8117,    and    S435  lbs. 

Thus  the  curve  A't,  obt, lined   by  iilottiuj,-  these  values,  shows  the  v.i- 
riation  of  tln'  iir.--<siire  in  (he  chainber  surrnun<lin);  the  wheel, 

22.  Large  Orifices  in  Vertical  Plane  Surfaces.^  1  l.i 
i-<-uiii"  jet  i^  .ipitiMNim  itel\'  of  tin-  siinic  scitiotial  fnnii  .is  tlic 
uritH-o.  .m.l  tlie  fliiiii  bl.iiiKMits  convor^a-  to  a  iiiiiiiimiin  section 
,1-,  in  tlie  case  of  simple  sliari)-eti;.,'e(l  orifices 

ut\  K.itittii^iilitr  Orljhr  .V\'^.  501.  — Let  /;,  /  I.e  tlie  upper 
,111(1    lower    ed^'es    of  a    larne    rectaii^nilar    orifice    of  breadth 


l.JRiJF   RFrT.-tSGn.-ln'   (iKIIICFS. 


79 


A,  ami    U-t    //,.   //.\)c   ihv   ikpths   ..t"   /;    and    /,    rc-in  rtivclv, 
below  the    tViT    -urtac-.:  at  .1        It    ii  lu'  tlu'  vcj.icit)-  uitli  wlnJi 

tlic  w-Ucr  rciulu'^   the   •■ntici-,  then  //  -=  ^__.,  i-  the  tall    ot   tree 

surface    wiiich    niu-,t    liave    l>een    exiiemle.l    in    ur.  i.liR  ini;    tlx- 
\el<)eit_\    //. 

ilence    //,  +  //  and  //,  +  //  arc   the   true   depth-   ..t   the 
edi^e-  /:'  and  /•"  bclow  the  -iurtaec  nf  still  water. 

Let    MX    he    the    niiiiinmni    'T    eiintraeted     -.eamn,     and 

assume  that  it  it  i^  a  rettani^le  ot' breadth  /'. 
Let  //.//,  be  tlic  depth-  ■>(  M  and  .\',  res])eetivel_\-.  beh.w 

tite  free  Mtrtace  .it  .  /. 
'riien    //    -.    //,  /i,Arlf  are'    the    true   tlepth-    ni  .1/  and    X 

behiw  the  surtaee  nf-tiU  water. 
/■irs/.       Let  the  tl'iw   lie   into  tlie  air,  the  oritlee  bcin^'  elear 
above  the  tail-uater  les'i-l.    V'vj,.    50. 

Consider    a    lamina   of  the    thud  at   th<'    -eetioii  .l/.\',  of  the 


i     : 


^    ''i  1 


^. 


S..-. 


li 


...i.  t- 


: I- 


Flu.    5( 


wi(ltl)    of  tlic   ■^eetion,  and  between   tin-  il'pth-    1    .\\n\  x  -\    lix 
below  the  surface  of  still  water. 

The  elementary  tliscliar^'o  <///,  in  this  lamina,  is 


liq  —  bifx  V2gx, 


8o  l..-iRGl-    RF.CTJSCjUl.AR   OKIHCHS. 

.mil  llR-rcforc  llic  L.tal  di^^liaryc  (J  .uro--  the  section  JAV  is 

II   ■■  _  (;,^^!if\. 


=      />  ♦  -,V  "''.J 


Put 

'llun 


/;,(//,4-//i^ -I //,  +  // J^i" 


s3. 


(1) 


The  c'lcffuiiiit  (  is  by  IV)  means  constant,  hnt  i-  tound  to 
vary  both  witli  the  liead  >  >\  water  and  als, ,  with  tlie  dimensions 
. it  the  oritiee,  and  i..tn  only  he  deternnneil  !)>■  e.\i)ernnent. 

SiiCiiii.  Let  tlie  orifice  he  jiartiall)  J'"iL,'.  5IJ  suhinerijed, 
and  let  //  ,  he  tlie  depth  between  the 

surface  ol  the  tail-race  water  and  the     <; f" 

free  surf  u  e  at  .  / . 

\\\  wli.it  ]irecedes,  tin:  dischari;e 
Q^  throu;^di  l.d.  the  portion  of  the 
orifice   clear   ah' >\  e   the    tail-race,    is 


II 


c;,  =  ".,/.'<-:.-(//,  .1  //)? 


//j 


(//,  i-//i^:.     (2> 


C,   *- 


i-'.\ery  fluid  fil.iinent  flows  thri '11:411 
the  jjortion  <,'l-  of  the  oritice  undt-r 
an  effective  heaii  //.,  -♦  //.  and  there- 
fore with  a  \elocit_\-  e(]ual  to 

Hence  tlie  dischart^c  (J.  throu',di  C/7' is 


Fir,.    51. 


.and  the  total  discliar^'c  Q  is  ('(jual  to  0^  -\-  Qy 


(3) 


lAKi.i:  m.cr.^M.n.'iK  okiucfs. 


81 


The  cocttkirnts  ,,.  .,  arc  t..  he  .ietcriiiMicd  by  c.xpcrmicr.., 


ami  if  c\  =  C,  = 


O  ^  a  +  (),  =  <■/,'  s  2c  r'  ;,//,  +  //■■'->//,  +  //^': 

~  L-3 

Tlni-d^    Let    the    nrificc    lu-    wliolly   siit)mcri;cd    (Im;-;.    52J. 
I'hcii  tin-  t'ltal  aiscliari;c  0  is  evidently 


l: 


CJr-,/M    2-i//_,  -   //,)   t    //:;-r//.      fS) 

<•  heillL;  a   idcftk  iriit  tu  he  deteriuiiud 
by  CNpeiiiiient. 

If  the  \elMeit\-  >  if  appf  lach.  //,  is 
-,utVi.ieiitl\  -niall  t..  he  ili-re,L;arded 
witliout  sensible  eni>r.  tl.^n  // —  o, 
and  equations  \  1  1.  4:,  and  1  5  .  rtsjiec- 
tiveK".  heennie 


I--;.,.  =12. 


.\  iiW 


11. 


Q  ^cHs  2^\U}\lI--^i 


'11: 


'7) 


(?:=.</;  \  2ci//,  -  //,)//,'. 


ill)  Circular  OnJii,.<.  -Let  V\'^.  53  lepn-ent  the  niiniu!.  ni 
section  nf  the  eiicular  jet  issuini,'  Irmn  a  circular  orifice. 

Let  2"  be  the  ani^le  siibteiuled  at  the  centre  b\-  the  \\\m\ 
lamina  between  the  depths  .1   and  x    •-  lix  below  the  surface  .it 

still  water. 

Let  r  he  the  racHus  (if  the  section  so  that  2/-  =  h,  —  h^ ,  //, 
and  //.,  bein-.  as  in  ui\.  the  depths  of  the  hi:^-hcst  and  lowest 
points  of  tlie  orifKc  below  the  free  surface  at  A. 


82  LAKGfi    CIRCULAR   ORIflCf-S. 

II.  ;is   t)of.irc.  i-,  the   lic.id   corrcsixiiKiini;  to  tlic  velocity  "f 
;ipproacli. 


( 


1  h,.  v^. 
'1  hen  the  area  of  the  himina  iiiider  ennsuieration 
=  2)-  -'ill  "  .  il\ . 
and  the  elementary  <li-^ehar,i;e.  ,/,/,  in  tlii-  lamina,  is 
(/(/  ^-  2y  sin  «  .  dx  S  2;^x. 

and  therefore 


dx  -  r  sin  f'  .  </". 


I  lence 


,/,/  =  2r-  sin-  w  v/  2,c(^   '  ',  -  >-  co--^ 


^  ^dH, 


and  the  total  disehar:4e  Q  is 


=  .,-M2.,y\in^^(''^t^^''       ;cos.>)%A.       (9) 


Ex.  The  free  surface  on  the  up-stream  side  is  5  ft.  and  on  the  down- 
stream side  I  (t.  above  the  sill  of  a  rectangular  sluice  12  It.  wide.  How 
much  must  llie  sluice  be  raised  to  i;ive  105.000  -als.  per  minute  ? 

10;. 000         „  , 

105.000  gals.  per.  min.  =  (,1^^  =  -f'O  cu,  h.  per  sec. 


NOTCHES   MND   IV t IKS. 


85 


Let  X  ft,  he  the  npcniiiK  ubov.-  the  m!1.  Tor  a  .iopth  ..f  •  ft.  above 
th.-  sill  the  -iischiirse  is  imder  a  constant  liead  of  5-1=4  It.  For  the 
reniainricr  of  the  oin'tiini;  the  (lischari,'e  takes  place  freely  through  a 
reitaiiuular  orifirc,  with  it^  upper  .ind  lower  boniularies  respectively 
(5  -  J)  li.  and  4  ft.  b.low  tiie  up-sireani  surface.     Then 


280  =  4-  12  ■  '  •  ♦  ''M  ■  4'  +  -    •  c 


Therrfore 


440  -  40(  5  -  .<  I' 
I  ;  -    rl'  ^  4. 


V'M 


=  2.48  ft. 


->J'  +   4'; 


23.    Notches   and  Weirs. — When  ai\   orifice  extends  up  to 
the  free-surface  level  it  hecoines  what  is  called  a  h./U/i. 


Fic.  54. 


Fic,.  ;■;. 


A  r>v/;-  is  r.  strucnire  over  which  tin-  water  llnws,  the  dis- 
c]iarL,'e  beiiiL:  in  tlie  same  conditions  a-  fir  a  notch,  and  i--  \ery 
useful  fir  t^au-in-  the  How  of  -mall  stic.ms,  the  .iniount  of 
water  supplied  to  hydraulic  motors,  etc. 

Ri\  fan  ■;■!//.!>■  .^■''''/'  "  ;f,^;.  —  The  dischar^^e  ina\- he  f.und 
In-  puttiuL;  //,     -  O. 

Thus  eciuation     i     becomes 


Q  =  " ,/.' » 2.i::(H.,  +  //)5  -  //^i 


do) 


If  the    \elocity  of   appi'oach    he    disregarded,    then    I f '^  O, 
and  the  hist  equation  he<,  onto 


Q  ^  "  c/i  V  2ir//}, 


(M) 


84 


lyEIRS. 


an.l  //   .-  the  (K'l'tli  t..  the  hcttein  ol  tlic   n..Uh  or  i..  the  crest 

of  tlic  w  cir. 

Croat  care  sliouKl  be  taken  in  ..btainin-  the  accurate  vahic 
,,f  //,.  A  hook  or  a  stilT  vertical  rod,  with  a  sharp  point,  may 
be  fixed,  at  a  suit.d)!-  distance  (?  to  S  ft.)  from  tlie  Lack  of 
the  weir,  with  tlie  point  on  a  level  with  the  crest  of  the  weir. 
The  tlunic  IS  then  filled  with  water  risin-  sli-htly  above  the 
cre.t  and  prodiicin-  a  capillary  elevation  of  the  surface  at  the 
]irint  The  water  is  nr)W  allowed  to  subside  until  the  eleva- 
tion i>  lurelv  perccptib'e.  when  a  hook-f;.iu-o  tChap.  Ill)i^ 
adjusted  .m.ra  rea.lin-  taken.  .\  second  readini;  is  taken  for 
any  required  dischar^'c  over  the  weir,  an.l  tiie  difference 
between  the   two  readin-s  is  the  depth.  //,.  --f  the  water  on 

the  crest. 

It  h.H.  been  found  that  the  discharge  (C^)  is  appreciably 
affected  by  vibration,  and  it  is  therefore  of  importance  that  the 
weir  should  be  nude  as  ri^id  as  possible.  Ihe  up-stream  face 
of  the  weir  i-  nearl\-  always  vertical  and  .it  ri-ht  an-les  to  the 
direction  of  (low. 

To  diminish    the   eiiect  ni    the  velocity 

ualcr— ection  in  the  flume  >-hould  be 

lar^a-  as  compared   with   the  seition 

(if    the   walerw.i)-  on   the    crest,    and 

the  ileptii  of  the  weir  shoulil  therefore 

he  at  least  /rivVr  the  depth  // ,  of  the 

water  on  the  cre-t. 

The    crest    should  be    horizontal 

and,   ^'ener.dly  spe.ikin^,   it  consists 

of  a  pl.ite  with  .i  bevelled  edj;e.   I'i^,'. 

54,   on   tile   up-stream   side,    or  nf    ,i 

thin  pl.itc.  iMK'.  55.  "^"  lli'^  '■"'•  ''''^'^'  '*1"'"''^'  ''^■■"    '"""  ''"^^ 

inner  ed^;e.  ...  1 

.\  ;v//W.Y  edt;c,    l" ik'     5^>.   'liminishes  tiu    discluir^'c  and 

should  be  avoided,  as  us  effect  is  uncertain 


if  .(pprn.ich, 


the 


\ 


tl 


H'EIRS. 


85 


The  lonf^'th  />  nf  the  i  rest  should  be  at  least  thfCf  times  the 

.lepth  //.. 

The  effective  s-ctional  area  of  the  water  flowini;  throu^^h  a 
rectan-\ilar  notch,  or  over  a  weir.  i->  Ic  -  than  />'//_,  because 
of  {a)  crest  contraction,  i^)  end  contraction.  1.  i  the  fall  of  the 
free  surface  towards  the  point  of  discharije. 

It  IS  reasonable  to  assume  that  the  diniinution  of  the  actual 
section. d  are.i,  /.'//,,  due  to  crest  contraction  ,ind  to  the  fall 
(.f  the    tree-surfice  level  i-    prnporti' .ilal  tn    the  width   /.'■>!    the 

openin;_;. 

S/iffriss,,/  U'./'-.  rr  W'rir  -H'itli.'tit  Ind  C<»Ur<i,0<'>is.—  ll 
■I  w^'ir  occuines  the  whole  width  ..t  the  stream,  or  ihime.  FIks- 
5;  and    59,  the   contracti^.n  at   e.ich    einl  i'.  wholl\   suppressed, 


fl- 


F 


vJ 


Vir..  57. 


Fio.  58. 


V. 


L 


Fio.  59- 


J 


.uidiii-t  contraction  only  takes  pl.ice.  ie..  tin-  fillini.,'  shoot 
ol  w.iter  1-  techued.  ui  l/iiikiuss  near  the  ue-t.  .\ir  must  be 
freely  admitteil  below  the  f.dlin^;  slicet.  as  otherwi-.e.i  p.iriial 
or  compute  v.iuiuin  wdl  he  produceil  .md  the  -heet  will  he 
dcpressod  or  will  adhere  to  the  face  of  the  weir,  while  the  dis- 
char^'e  C'  will  bo  vory  sensibly  modified.  Iranci-  effected  the 
frte  adndssion  of  air  and  also  prevented  the  l.it.  r,d  sproadin^,' 
ot  ih.'  sheet,  after  leavin-;  the  crest.  h\-  pi  oloti-m;;  the  upper 
portions  of  the  flume  sides  a  short  distance  beyond  the  weir, 
Fi^'.  58.  'Ihe  discharge  was  thereby  diminished  by  about 
.4  per  cent. 


86 


Hi:  IKS. 


ll,/r  7.7///  /:>ii^  Contractions. — These  contractions  occur 
wlirii  the  sides  of  tin-  weir,  or  notcli,  l-'icjs.  ^^o  anil  T)  i ,  are  at 
a  distance  from  the   sides  of  the   channel,  and  they  have   the 


x 


-—-II- 


M 


._B ^ 


M.      Jfcfc.J 


Fi'i.  60. 


Fig.  t)i. 


ctTcct  of  iliminishin:^'^  the  dischari^c  The  >  ontr.iction  is  cow- 
phti.  i.e..  .IS  ^re.it  .i--  it  i  ,in  lie  iiKule.  when  the  dist.iiicc  of  the 
weir  side  from  the  ili.mnel  side  is  not  less  than  about  the 
depth   //.. 

(  )tlier  tl)in.i^s  bein^  cqu.il.  the  i  untraeti^in  and  its  clfect 
upon  the  discliarj;e  increase  uith  // . .  1  In-  etle^  i  ■  .1  end  mn- 
tr.ictions  is  almost  inappreci. diU  .ind  ni,i\  he  disre-.inled  when 
the  length   /.'  of  the  crest  i^  not  lc->--  tli.iil  .iln'nt  //,  ;    tnit  .is  the 

I'.in'iiiiislif-.     the    effect    r.ipi<ll)-    increasi's.       j-'r.mcis 


l.ltK 


//. 


iMiind    th.it    tin    dis,  har;;e  tni   a  weir   with    jurlei  t    end    ccntr.ic- 
tions  ,iiid  in  whuh  /■        1//.,  was  diminished  <»  per  cent. 

In  his  Lowell  weir  experiments  he  also  found  that,  for 
depths  II,  A  //over  the  crest  var\in^;  from  \  ins.  to  24  ins., 
and  for  widths  /.'  not  less  than  three  times  the  de]>th.  .1  jier- 
fect  end  conlr.uliMii  li.id  the  efTect  of  diminishing,'  the  width  nf 
the   fluid   section   1)>    an   amount  aiipreiximately  ecjual   t'>   "iie- 

//.  4-  // 

tenth   "f    the    depth,   or ,   so  th.lt  the  ellective    width    = 

'  10 

/A  1  // 
~"       10 
Thus,  if  there   are  ;/  end  contractions,  the  effective   width 

li  ^         (//j -)-  // I.    and   the   ecjuation   tjivin^;   the    dl■.char^,'e 


li 


IVEIRS. 


87 


becomes 


According  tn    Kr.uuis  the   average  value   of  ,   .n  th>s  .,,ua- 
"""rhenl-  ,^  =  3.33.  vv.y  nearly,  and  thereO.re 


Q 


th 


In    experiments    carried   out   by    luUy    and    Stoarns   ., 
ssed  weirs,  as  .lescr.hed  above.  tl,e  total  vanat.on  ,n  the 


supjire 


.X  of  the    coefficient  wa.   0.und    to    ,,.    about    ^n--;t 


The  depths  //.  were  mc 


isured  t.  tl.  trom  the  weir,  .ind  tor  v. dues 


of/Aexceed>nu.07ft.  thevdeduc.-dth     tormula 

Q  =  /ij,.\\//.}    t-  -0*^7 )• 

,n  vv  hu  h  the  velo,  ,ty  of  a,.proaU,  .s  d.sre;^arded 

\„„uan..-  nuvbe  um,!,  ,0,-  the  velout  v  of  ,.ppro,u  h  In 
substUutin,  ....  //.  tin  expression  //..-.i//  ■-^"l^'^^' 
Ktclevand  >t.un..  but  //,  ^  -J//  accnrdm,  to  Hanulto,, 
Smtth,   Ir.  who  bases   luscon.luMon.   upon   a. onMunv..|o. 

f  l.>..li>x-    Hid    sttMrnN  \\ith  those   o|    1  lanci-- 
the  experiments  ot    1  tcle\  .um 

.iiiil  others. 


/r  .  n 


.1 1) 


inu^l   De 


If  the  weir  h.is  ;/  end  mntr.u  tions,  /■' 

subsftutcd   tor    /;.  and   allowance   ts   made   ,or   ^'-  vel-^y;;^ 
1...  .nbstitutin.j   for   //,  th-  expression   /A +2.05 

to 


substituted  .or  /;.  and  allowance  ,s  ma.e  ,or  -^-••-  - 
approach  by  substituting  for  /A  th.  expression  /A +•  05 /A 
^ccordin,  to  l--teley  and   Stearns,  or  //,  +  .4//.  -cordin,  to 

Hamilton  Smitli.  jr. 

Hunkingand  ll.ut  ..ue  the  tormula 


88 


III  \UiilIi  u  is  \-t.r\'  ncai'U'  =  i  -\ ."    ,  wlicrc 


SUHMEKGliD    H'EIRS. 
sc'Ctiiin.il  area  of  wattruav 


/;  -  // 


/'/. 


lo 
liaziii  Ljivcs  tlic  torinula 

in  wliii'Ii,  iftlic  velocity  nf  .ipproatli  is  disrci^rar.K-d, 

.OO9.S4 


c  =  .40;  + 


//. 


but  if  allnuaiKc  is  t.i  he  made  for  the  \ilorit\-  nf  approach, 


.CX39,S4\    » 


'•  =  1405 -'-';;^j ;  i  -{--s.h 


II.,  +  x)   )  • 


X  bc-itlt;  tile  lu'i^dlt  r)f  till'   wcil. 

Ha/in  considers  that,  with  suppressed  ui  irs.  as  alrcad)- 
described  and  uhiili  are  not  \  er\  low,  the  n-^ait-,  obtained 
with  this  CO!  tfu  ient  are  acciirati'  witliiii    !   per  cent 

Suinntigi,/,  ,■!■  />;•,»?,■;/<</,  Iiaiiis  {or  \\\i>s).  —  \x\  these  the 
surface  nf  the  tail-race  water  rises  above  tlie   top  of  the  dam. 


Fi.;.  62. 


I'Mfj;.  62.      It    riui>   be   assumed  (ii.it  iietween   a  and  /'■   the  tlou 
is  as  over  the  crest  of  a  w<  u,  tin    depth  ol   u.iter  on    the  crest 


INCLINED   U'lURS. 


89 


l)L'mL:  /'',  —  //,  and  thai  between  /'  ami  <"  the  flow  i-;  equiva- 
lent tn  that  throut^h  a  siibmcrtjcd  orifice  under  a  constant  liead 
H,-\-ll.  liiiive.  if  //  is  thi'  dejith  of  the  top  of  the  dam 
heh'u  the  surface  of  the  tailwater,  aiui  if  <  is  the  coefficient  of 
di^iharL^e  both  t'or  the  flow  betwcer.  a  and  /'  and  also  that 
between  /'  and  < , 


The    loUouni^    table   ^dve<  approximate  values   ot   ,   torre- 

'jpondin:^    to    dillerent    \alues    dt    the    ratio  .  as 

deduced  from  exiJeriments  carried  <iut  In'  I-'ramis,  the  head 
over  the  crest  varNini,'  from  i  to  2.;i2  ft  ,  and  In-  l^'teley  and 
Stecw-ns,  the  head  \ar\in;.[  from  .325  U-  .S15  ft.: 

*^'?  CorrjspondinE  Valurs  <>f  <  as  ileilucrd  from  the  f«p< nmciUs  ol 
— —■  Krami^.  Kicley  and  ;>ti-.irns. 

.05 (i23  t'l  .'132  

.to 620   "  .631)  .62510.635 

.20 ("O    '■  .625  .618    ■■  .623 

.30 508    "  .615  .600    "  .6X0 

.40 5S6    "  .610  .590    "   .60(5 

.50 585  '•.607  .5S5  •'.595 

.60 585  •'.607  .583  "593 

.70 5S5  •■  -to;  .5S0  "  .590 

.80 585  ■•  .607  .581  "  .59' 

go      .590      "     .'"X) 

.95 .61"   ■•    615 

(  J'raulii  inf.) 

bhlithii  H'c/V-s-.  — If  the  up-stream  face  of  a  weir,  instead  of 
bein^  verti.  al,  is  inclincii  up-stroam.  I-"i^'.  ^3,  the  discharj,'e  is 
diminisluM;.  the  d-pre-^sion  of  the  upper  surface  of  the  fallini; 
slicpt  of  water  commeiues  nc.ir  tiie  crest,  while  the  lower  sur- 
face rises  hinlier.  above  the  crest,  and  moves  backwards. 

It  the  face  is  inclined  down-stream,  Fifj.  64,  the  discharge 
is  increased,  the  depression  of  the  iipjxr  •surface  commences  at 
a  point   farther   from   the  i  re>.t  tlian  when   the   f',u  <■  is  vertical, 


Il 


90 


cmCL'LAR    NOTCH. 


while   the   lower    surface   heCMiiies    nmre  flattened    and    moves 
a\va\'  irorn  the  weir. 

Values  of  the  coefficient  'A  di^cli,uL;e  for  inclined  veirs  h.ive 
been  deiluced  b\-  Ha/in  and  arc  L,Mven  in  a  sub-^ei]uent  article. 


Fig.  63. 


Fic.  t)4. 


The  dischari^e  is  incr.  ,i-^ed  b\- roundini,'  the  u])-stream  edi^e 
<,f  the  weir. 

t  iraJ'ir  .\'-n/i.~h\  e-juation  (9;,.  Art.  :_\  l>ut  //,  ^  O  .md 

//,  =   2r.       'I  hell 

Q  =  21"  S  2,1:    I     -in-  f^ill  -\-  21  Mu-  ^  j',/^'. 


and  if  tin   veh.cily  of  appmach  be  disregarded,  ->..  that  //  -  O. 


d» 


=  ♦  'V 


/       J  sin      -  sin  --  +sin   .,  J-/'* 


64    ^ 
15        * 

1  X  1  \  a.mi  «ith  a  rectangular  notch  6  (i.  wide  is  formed  across  a 
channel;  and  the  d.pth  of  the  water  over  the  sill  Is  12  ins.  Find  the 
quantity  of  flow  when  the  notch  has  (..»  no  side  contraction;  {f>)  one 
side  contraction  ;  (<)  two  side  contractions. 


EXAMrLliS 


9' 


I)isP-},nirfilng  tlie  i  elocity  of  approach,  and  assanimf,'  the  i  oefficieiil 
of  cliscliarj,'e  to  be  the  same  in  each  case,  viz.,  k. 


(}    _    ^1     A  .  ^'64  .  il  .  6  =  20  cu.  ft.  per  sec. 
3  ■  « 

(\  =    "  .  '   .  t  14,  15(6  -  ,'o)  =  n,\  cu.  ft.  per  sec. 

3     i^ 

(?,=  ■.'.■»  '.4  .  in6  -  ^\  ^  19s  <•"■  "■  l'^'''  "<■'-■ 
3     S 


(*) 

Ex  '  400  ru.  ft.  of  water  per  second  are  conveyed  by  a  <  lianiiel  ..1 
renan-ular  section  25  ft  wide,  when  tlie  water  nins  4  ft-  deep.  Find 
the  h<"i«ht  of  a  dam  built  across  the  channel  whicli  will  increase  tin- 
depth  50  per  cent,  taking'  into  acount  the  velocity  of  approach. 

400  S 

The  velocity  of  approach  -=  ,5   j<  c  ^    3  "■  ^^^  *"■ 

The  Cdrrrespondiiii;  head  =  ^^--  -    fil- 
let  1  ft.  b<-  the  heii;ht  of  the  dam. 

First,   .\ssume   that  the  .lain  is  not  <Acr:<.«,,/.  ..e.,  that   its  crest  rises 
above  the  water-surface  on  the  down-stream  side.     Then 


400 


♦  '64',  (6  —  .1   +  J)'  -  iji 


,s-5 


A 


or 

3,„,    r  =  3.25  ft.,  which  IS  /«,t  than  4  ft.,  and   therefore  the  aSMimptinn 
liiat  the  dam  is  not  .frowned  is  incorrect. 

Secomi.  .Assuming'  that  the  dam  is  drowned,  the  .iischarne  n.jw  takes 
place  under  a  conM.mt  head  .)f  (2  +  J)  ft,  for  a  depth  of  (4  -  -«•)  «t-.  and 
as  over  a  weir  f.ir  .1  depih  <.f  2  ft.     Then 

400=  ^^514-  -♦«>4(^  +1,*+  -^.|.2s.V64.;.2^ir     (I)':- 

or  i-'798       (4       ^)i25)*- 

,j„^  ^  ^  3, 88  ft.,  which  is  L'ss  than  4  ft.,  an.l  therefore  llic  assumption 
of  a  dr.)wned  .lam  is  orrect,  ,  .  1   , 

Ex.  3.   If   «•  1-^  ih^  '''^^1"''   "'   "■■'^*-"''  "^'-'''  '^'"^^^        ^  rectangular 

notch,  then,  disregarding  the  vcl..city  of  approach, 


ill 


92 


TRIASCAJLAK   SiVK.H. 


Let  «/<;j  be  the  chan^jc   in   Uic  discharge  corresijondini,'  to  a  ihaiige 
</r  ill  the  deiith  mi  thr  ~il..      I  iifri 


,i<2 


:/;  V2g.     ; 


Henre 


"V  _  3  '''■' 


Thus  a  change  f)f  6  per  cent  in  llic  discharge  corresponds  to  a  chaiK' 
of  4  per  cent  in  the  sill  depth,  and  a  change  of  lo  per  cent  in  this  dcplii 
rtjrrcsponds  to  a  change  of  15  per  cent  in  the  discharge. 

24.  Triangular  Notch.  —  l)i-ii,;^anl  tin-  velocity  c^f  ai)proai.li 
.111(1  let  /)'  t)c  the  \U(ltli  of  the  tree  surt.ice. 

As  before,   consider   .1  l.iniitia   "t 
tlui<i     between     the     depths    .1      ,nid  . g > 

The  .'ire.i  of  the  l.imin.i 


//J 


y/..  -  xuix. 


Fig.  65. 


.-ind  the  disihari'e  in  tlii^  l.ii-.in.i  is 


/.' 


,itj   =z  I         [If.,   —  x)tix  ^'2jfX. 


Hence  the  tot.il  dischar_L,'e  <J  is 


=  i ^j  S  2,^-   /       (//.r'  -  x^),/x 

-    '^. .  A' ♦'2.i,'-//J  eu.  ft.  per  sec.       .      (14^ 


r  is  a  coefficient  introduced  to  allow  for  contraction,  etc., 
and  I'rofcssor  James  Thomson  gives  .617  as  its  mean  value  for 

.1  sharp-edi;ed  trian^piLir  notcli. 


USAMPLF.S. 


93 


Nou-  tlu;   ratio    ''^     i^   cnn-^tant   in    ,i    trian-ul.ir   nouh   and 

vari..  in  a  rcctan-^ular  n..uh.  Hence  Th.Mnv.n  mtcrml  an,l 
shoucd  bv  experiment  that  the  value  o!  ,  .-.  nmre  un,t,.rm  lor 
trian-ular  than  f-r  rcetan-ulav  nntel,.-.  and  tiieret.-re  al>w  the 
f.  .rnier  nui-t  -i\e  nmn-  accurate  results. 

If  tlie  n^w  i-  ihrou-h  a  Qo"  notcli,   />'  -  2II ...  and 

()  =   '"^  ,  \  -•//'-■  -^  "Mil''  ^■"-  '"t-  I"-''  '"'''■••  :M'P'""^'">'^^^''>'- 
15         "•■•     ' 


,   heini;  .'ii  7  and 


=  I  ;S.  ^S;/Aj  cu.  It.   per  inin., 
3-- '/"'''■ 


Fx    I      -V  reservoir  cliscluirj;-s  tiiroUKh  a  sharp-e.lt;c  tr.an-ular  notch, 
and  in/  sees,  the  dcj.th  ..f  the  water  in  the  notch  (alls  fnan  //  tt.  to  .  fl. 

Let  V  be  tlie  sectional  area  of  tiie  reservoir  corresponding  to  the  .. 
!,    depth-  kl  ;//.i   he  the  width  of  the  free  surface  on  the  notch  corre 
.ponding  to  the  .1-  fl.  depth,   m  bcin;;  a  nmr.erical  coctiicient  depending; 
upon  the  notch  ani;le. 

Then,  since  the  water  sinks  </-r  ft.  in  <//  .sees.. 

5.  ,/.i  =  discharge  from  reservoir  in  <//  sees. 

=  amount  liowmi:;  throuf^li  notcli  \n  dt  sees. 

=     ■*     S' Zi^C  .  "M"  .  tit. 


Jl  =  - 


Iv'>' 


.  Jx. 


Heiu  e  ///^  ///"<■  /«  s.vs.  n.  which  the  d.-ptli  falls  from  H  ft.  to  x  ft. 


I-  i»"       -J 

4   \   2^'C»lJ, 


If  the  horl/.ont.ii  ■^(■ctlonal  iirea  .^"  is  c<mstant. 

For  a  'ff  notch  m  =  z.  and  takinK;^'  =  32  and  r  =  |, 

'^x\    -J/i} 


tlie  tim<-  in  sf,- 


94 


iih'().-iii-<:Ki:sri;i>  m-7:7A'5. 


The  limr  herumcs  iiiliuiU'  ulu'ii  i  _  u.  uiiuh  indicates  tluit  the  tl'iw 
timi!iiishr>  iiiiieliiiitL'ly  will)  the  liepth  in  the  ncitch. 

K\.  2.  I'mii  tne  dischar^je  in  j,'all(ins  per  niimiti-  lliMiiij;li  a  90°  sUar[)- 
e(!ge  iiDtcli  whi'n  the  water  runs  4  ft.  deep,  ll  the  reservoir  .supplyinir 
tlie  water  lia,-.  a  eonstaiu  hunzntital  sectiona!  area  of  So.ooo  sq.  ft.,  in 
what  lime  will  the  level  sink  3  ft,  ^ 


f,^'"^-,s 


//!/•  inui' 


.4'  -T  ,S5J  eii.  ft.  per  sec.  ::   Sfi  ^  Ci^   v  60  fj.ds.  per  ruin. 
=  32.000  .1,'als.  per  mill. 


.Soooo 


(.- ■) 


17.500  sees.  =  458  liour> 


25.   Broad-crested  Weir.  — Let  I'it,.  C)C,  represent  a  strcani 
llouiii^  ii\ri'  .1  lifMil-crcstcd  Weir.      (  )ii  the  up-str'.'am  side  the 


Flc.  66. 


free  surface  I. ill-  iVmii  .1  tn  /,'.  \-\>r  .I'di-taiiee  /IP  cm  the  crt'st 
the  Ihiid  til.imcnts  ,ire  -eiisibly  rectilinear  ami  parallel;  the 
inner  ed^re  iif  the  crest  is  nmndcd  so  as  to  prevent  crest  con- 
traction 

Coiisidei  .1  t'll.iment  iii',  the  jxiint  ./  heiiiL;  t.ikcn  in  ,\  ]Mi"t 
of  the  stre.am  where  the  \el<-cit_\- of  flow  is  so  small  th.it  it  ina_\' 
be  ilisrcL;ardcd  witliout  sensible  error. 

Let  A  he  the  thickness  ,)/.\' of  the  stream  ,it  /'. 

Let  the  hoii/Liital  pi. me  throii^di  .\' he  the  d.itinr,  i>hine. 

Let  .;, .   :  be  the  depths  below  the  free    -inT.ice  of,/  .md  A. 

Let  //,  be  tile  elevation  of  ^z  abo\e  datum. 


BR().-in-(:Kl:S'l  f.K    li'l  /^'^. 


95 


I.t-'t/',,,  f  ,  />  be  tlic  atiunsplKiu   pri's^urc  ami  the  prcs-.iirc-. 
at  (I  and  /'. 

I.ct  :■  be  tin-  velocity  <')f  flow  at  /'. 
riieii.  b\-  Hernoviilli's  thenrein. 


*,+{:  =  A -.+^_;;. 


But 


P, 


A. 


tlierefore 


ami  heiiee 


.,+.,+f;  =  *-. +  .+{;  +  ;. 


2.^ 


^/,,  +  ,,_A  =  //,-A. 


//,  bein<;  the  .Kutli  of  the  er.        .f  tlie  weir  bel'.u'  the  -^urf.ice  of 
still  uater. 

Thus,  it'  A'  be  the  width  of  the  weir,  the-  disetiar^e  (J  \> 


Q  =  j;x  \  2xi  //.  -  ^ 


(16) 


I-"roni    this    eiiuati'in    it   ai)iiiais   that    <J    i-    nil  both    uht  ii 

A  —  o   and  uhen   A  =  //,        Hence   tliere    nui-t  be  some  value 

of  A  hetivcen  n  and  //,  f  .r  which  0  is  ,i  niaxinuini.  1  l''s  \alue 
nia\'  be  found  b_\    [lultniL; 


,/Q  =.  o  --.  /.'  ♦  2c(  i'//,  -  A.  -  ^  -^ 


//        A 


dX, 


and  therefore 


3     ' 


9<>  HKOAD-CRESTED   Ui.lKS. 

aiul  tlu'  expression  for  tlie  (lischari;e  becomes 


wliich  is  the  maxiTUiin  iliscli.ir;^'^e  tor  the  t;iveii  conditions. 

l-',.\perinient  slious  tii.it  the  more   correct  \Mlue  tor  tlie  liis- 
clKif'e  is 


C?  =  .35/'*2^/A". 


(i8^ 


If  tlie  water  apprnaclies   the  weir  with    an    appri'ciable  velocit}' 
;/,  corrcsnondinLT  to  the  liead  //,  so  that  —  //.  then 


'-^    II.  -\-    II 


A, 


and 


Q  =  .:,-.l!  \  2,::ll,:-  /I' 


This  forniuhi  ai;rees  with  tlie  ordin.ir}'  expression  for  the 
t!iscliar_Lje  <)\er  a  weir  as  L;iven  1>\   i-qnation     ii    ,  if  <■  =  .525. 

It  iiULjht  be  inferred  that  lor  broad-cri '^ted  weir^  and  larj^e 
tiiasonr_\- sluice-o[)enini;s  the  discliart;e  should  be  tleterniined 
b\-  means  of  equation  ;l8;  ratlier  than  b\-  the  ordinar)'  ueir 
formula.  \iz.,  cijuation  (ll). 

It  must  be  remembered,  however,  that  in  deducintj  equa- 
tion 17  ,  frictional  resistances  have  been  disrcj^arded  and  the 
;4ratiiitous  assunijUion  lias  Vjeen  m.ide  that  the  stream  adjusts 
itself  to  a  thickness  /  which  will  L,Mve  ;i  maximum  discliari,'e. 
Tile  theory  is  therefore  incomplete. 

The  ilischari^e  o\er  a  sharp-crested  weir  is  sensibh'  the 
same  as  that  over  a  weir  witii  an  apron,  .i--  m  1  iL,^  66,  so  lon^; 
as  the  depth  of  the  w.itiT  on  the  crest  is  not  less  th.in  .diout 
15    ins.,    l.)ut   below   this    limit,  tin-    discliar<^'e   o\-er    tin-    apron 


RF'irRrOIK  St. VICES. 


97 


rapi(il\-  (liminisliLS  with  the  ('cjith.  Fnr  example,  the  dis- 
thai'Lje  over  ,i  shai  p-crestcd  urir  is  ai)prn\-imately  double  that 
over  a  weir  with  an  apron  when  the  depth  is  about  I  in.,  is 
20  ])er  cent  .^'reater  when  the  depth  i>  0  ins.,  .uid  10  per  cent 
•greater  when  the  dejith  is  1  2  ins. 

26.  Reservoir  Sluices.-  I'he  water  IImws  into  the  rcceisin.t; 
channel  either  fiecly,  as  in  Fig.  O;,  or  under  water,  as  in 
Fi-.  08. 


Fi.;.  67. 


In  the  first  case,  the  stream-lines  converge  to  a  contracted 


Fir..  63. 


section,  am!  between  the  sluice  and  a  certain  section  />/:  tliere 
is  a  sudden  swell,  the  height  (.t-'\ell  being  given  b\- 


-.  J 


BC 


and  :    ,  :•,  being  the  velocities   of  flow   across  the  contr.icted 
section  and  the  section  at  /)/:. 

Let  .i    ,  .! ,  be  the   area>   of  the  sluice  and   section  at  DE, 
and  let  nA.^  —  A..     Then 


98 


RFSrK  K(  )lh'  SI.  LICl-S. 


i-   bcins  the  coefficient  of  cnntnution. 

Huh  the  swell  uiU  In-  f"un.l  tn  he  further  fmm  or  nearer 
the  shiice.  accor.lin-  as  the  .lifference  betueer.  the  deptli.  ot 
the  stream  anil  the  sluice  is    ,•  or  <-   AY  . 

If  A,  is  the  coefficient  "f  h\.hauhc  re>istance.  tluii 


{^+u-:^  =  ^'''> 


an.l  r,   mav  he  .i  nr  even  greater:  hui  ,f  the   sluice  cd^es  arc 
smoot'hea  an.l  rounded  m.  that  /,  can  he  disre-ard.  d.  then 


an<l  therefore  .1/:  -  A'i    -  -I(    —   ^  ,.■ 

'.  is  assumed  that  the  water  in  th<'  reservnir  retains  the 
same  level,  bat  where  the  flow  connnences  there  is  a  depres- 
sion in  the  surface  due  to  the  velocity  of  (l.-w.  and  the  amount 
of  this  deiiression  should  be  deducted  fmni  the  f.tal  head. 

When  th.    b:i.  kuat.r  ri^e-  ,d).>vc  the  sluice,  as  in  Fi^-  f>^< 

AC  =  head  required  t..  produce  .-,    j   lu.i.l  "  lost  in  shock  " 


and  AC  increases  with  h.  i.e.,  as 
with  /}... 


,r  diminishes  as  cnmjjarcd 


mm-  (U7:A'    HF.iRS   {li.4/.IN). 


99 


27.  BaziD's  Flow  Over  Weits.  —This  article  is  tlic  i'^suiik-  of  B.izin's  val- 
uable i>a|KTS  oil  tins  subject  [)ublisiit;<l  In  tiic  .ln>i,t/,-\  r/ci  /'outs  tt 
Chiiuisves.  Tlif  svinbols  ari-  <  1kiiii;('i1  I'l  1  urrcspoiid  with  llic  prcc  cdiiiL; 
articles  ul  the-  present  chapter. 

Let  c, .  /)', .  //,  be  ilu;  coelliciiiit,  length  of  (rest,  and  lieail  over  crest 
for  a  stand, ird  weir. 

Let  f,  Li,  11,  l>e  the  i:orrespon<liiit;  svinbols  for  an  e.xperinieiilal  Wf;ir. 

Tlieii,  disregaidinij  tlic  vel<j<  ity  of  approach, 


,  ,/.'.//J  ♦  2v  =  i>  = .  ///a'  V  ^. 


and 


ExpcriniiMit'-  with  the  >l,iiid,ird   weir  i,'ivi'  the  v.ihie  of..,  tlie  ratm 

'  is  usually  tt.'.'//i,  and   the   r.ilio    ..'is  found   by    observation.      Hence 

llic  va'    2  (if  <  can  be  at  once  calculated. 

In  pr.ictice  it  seems  irnpossilile,  with  the  d.ita  at  present  available,  to 
make  a  rational  sileclion  of  the  proper  value  of  <.  which  varies  between 
wide  limits  and  is  alfeited  not  onlv  l>y  the  form  of  the  weir  l)ut  by 
other  conditions,  ainonfjst  which  may  be  enumer.ited  the  following;  :  — 

(u>  The  Tr/cf//)' ly  <!//» (»<((//,  whiih  c, mi, ot  be  disregarded  when  the 

weir  is  of  small  hei^iit. 
(/•»  The  luii^ht  of  the  weir. 
(<)  The  iTisI  tOHtratlii'ii.  which  rlepends  l)oth  upon  the  hej^;ht  fif  the- 

weir  and  the  form  of  the  crest. 
{,!)  The  end  lOiilriUlions.  which  have  a  considerable  influence  whcrr 

the  weirs  arc  of  comparatively  small    width,  but  are   not  ot 

so  much  importance  when  the  weirs  are  lonij. 
if)  'T\\c/iirm  of  tlif  mippf.  which  niiiy  vary  considerably,  iind  which 

in  every  case  should  be  the  subject  of  a  c, ireful  investi^^ation. 

Sliitrfi-fmuii  Wftt  (Ki;;s.  54.  55).— The  simplest  and  best  defined 
t  .lo)'.  and  une  which  a'.mitK  of  an  exact  determination  of  the  coefficient 
of  ki  schargc  c,  is  that  of  a  fice  nap|ie  (or  sheet),  the  sheet  of  water  flow- 
ing over  the  weir  without  end  (ontrnction,  and  with  its  lower  as  well 
as  upi>er  surface  fully  exposed  to  atmospheric  pressure.  .MIow.inee 
may  be  made  for  the  iiiduenre  iiixin  the  dittcharije  of  the  velocity  nl 
approach,   «.   by    substituting  lor   the    head.  /A  ,  over  th':  crest  111  the 

discliur^'e  formula,  the  expression //,  4    ■>    ,    ,  <i  being  u  coefficient  wliicli 


has  n  atieen  aciuralcly  <l.neriiimra.      1  nu^ 


u  heint;  the  moditied  value  ol  c. 

Bat   -"'    is  very  small,  rarely  exceeding  ,i  tew  centmictors,  and  ihcic- 

fore,  approximately. 


q--rni;n^  syr{\  +  ■]"2^/7;)- 


Let  X  be  tin-  heiulit  '■!  the  w<ir.     Th.-n 

ulUlh  ^  .11  =  >J  -./•'♦  Ja  •//.^ 
and  therolDre 


2-//.      '    l///  +  .rj  ■ 


Hence,  puttini;    A'  -    \>ii\ 


n  ^  )i     I  + 


*■(/;.,)■;-'•■•-"■' 


30  that 


\UTM^  has  de.lMred  the  values  ..f  ■«.  A.  .uui  r.  I.y  .  omi>.,r.a,ve  cxpor,- 
nent*  on  tive  weirs  of  different  lieiKhls. 

■I  ind  A'  are  not  constant,  hut  their  iikm..  v.ilues  arc  \  and  .55 
respectively.     The  coefftnent  ,,  slowly  dmunishe*  as   the  head   /<   .1- 

crcases. 

Thus  _  _  . 

for  heads  =  o-^.os    o".io    o".2o    o'Mo    o"'.40    o».5o 

the  correspond.nK  values  of  «-    .44«       -43^       ^^i       -417       ^U      •4«J 

For  values  of  //.  -•  o-.io.  it  is  sull.riently  ac  urate  to  take 

.003 
^  =  .405+^-. 


H.Oir    ()yi:K    HhlR^   <HA/.IN\. 


101 


and  therefore 


SI  I  tliat 


(ieneralK,  fur  values  <i(  //j  between  o"'.io  uiul  o".30,  "  iiiuy  be  made 
e<iii,il  to  .4:5.  aiul,  takiiij,'  A'  =  .5, 


a  suitable  form   for  iir.ictical  u^e  when  errors  of  2  to  3  per  <  tin  ..re  n  )t 
too  lar(»e  to  be  of  itnpurlance. 

The  absolute  vahies  (jf  c  havinj;  been  found  for  a  sharp-crested  weir 
with  a  fiec  nap|)e  and  a  vertual  face  on  the  up-stream  skU'.  it  does  nut 
(<j11()w  that  the  same  method  shmild  l>e  adopted  to  determine  the'  cor- 
respoiuimi;  roelhcMents  for  other  forms  of  weir.  In  fait,  if  i'  is  the  coef- 
ficient for  anv  other  ).;iven  weir,  when  the  head  over  the  (Te-.t  is  the 
same,  the  inthience  of  the  vehicily  of  approach  may  he  largely  eliminated 

by    flndint;    the    ratio  The  ratio  corrrspondinp  to  two  dilfeietit  in- 

clinations   IS  sensibly   rfinstaiil    for   all    heads,  and    the   followinjj  tabic 
gives  the  values  of       for  varying  f.i(e-slo|M.-s  ;  — 


Foi  an  iip-stu.im  (.11  e-slope  of  i  hor.  !o  1  vci: 

_,  ..       3      ••    ,. 

,  ..       ^     ,. 

vertical   face 

For  a  down-stream  face-slope  o(  1  hor.  to  3  vert. 

:  ■■      1    " 

1  ■•       1 


■'>3 

=  -94 

=  ../. 

^  1.00 
1.04 

=  1.07 

=  1. 10 

=  1.12 

—  I  .<:•<; 


It  m.iy  he  iioteil  that  the  coefTlcietit  loi  lali  1  j^i.nhiaiiv  mh  ri.isrs 
from  .93.  cotresponilin^j  to  a  slope  of  45*  on  the  up-»ilreiiiii  »idc.  to  i.U, 
correspondinK  to  a  slojie  of  about  30'  on  the  down-ttreum  side. 


102 


iioir  oihR  li'hifx-s    n-1/.is 


When  till-  air  cannot  pass  iiuderncatli  tlii'  sliect  of  wa'.ei  tluvving  over 
the  crest,  the  iiajipe  cither  encloses  a  volume  of  air  at  less  tlian  the 
atniuspheric  pressure  anil   is  ih-presnii.  Fit;,  (jy,  or  the  an  is  entirely  ex- 


I'lii.  6'j^  ^Depicsso'l  Nappe. 


I'l'..  7'i  -Drowni'il   Nappe. 
>l,\llll   nr  ,h'>7^'>i,;i.  I'ii;.  70.      The 


!  !ii(1c(i  and  the  nappe  is  wetted 
iattei  coiKiilion.  when  the  nappe  «  iosc  an  edilyini;  ni.issnf  llnid,  i;ivs 
a  more  unilorni  motion,  as  the  pressure  of  an  enclosed  volume  uf  air 
may  v.iry  from  tiie  ai(:i<lental  admission  of  new  air.  The  di-<iiari;e  is 
sii^lhtly  greater  than  with  llie  free  nap)pe,  and  may  he  imieased  almost 
10  per  cent  when  the  nappe  is  on  tlie  point  of  hcinj;  drowned.  So  !on;4 
as  the  head  exi  eciis  a  certain  limit,  the  iiafipe  will  not  be  m  contact  with 
the  weir  f.ii  c.  The  drowneil  nappe  may  be  ether  indepen  lent  ..f  ..r 
influenced  hv  the  down-stream  level  according;  as  the  rur-  produceij 
beyond  the  nappe  is  at  a  distance  from  the  foot  of  the  nai)i>e  m  partially 
enchises  the  foot. 

A'/.ir  at  ,1  Iht.ir.:  ,•  f),>m  t/'ii-  /'.'.'/  <>f  tfif  .\\>f'f>f.      In  tiii>  1  .ise 


.878  -t-  .i:.S 


/// 


hut  the  max.  value  of  -      cannot  ex<eed  2J,  as  the  (/ro^'//*-./ c<  edition  no 

longer  holds  when  //,  •    '  i.      The  v.iliie  of'  .  corresponding  t.i  this  maxi- 

nmm,  is  t.2,  and  if //,=.»,  the  cocfTKienls  ,' and  <  arc  sensibly  the  same. 
Applying  this  '  trimila  to  wi-irs  of  <litlercnt  heights,  it  is  found  that  tin- 
absolute  values  of  the  cociricieni-  -f  'I'-iliarye  ar.-  ^ii.^ib'v  yiv.-n  by 
the  formula 


.'  =  .47  +  .00;  j 


//.' 


Ris,-  EnclouHg  lit,-  l',iot  ,\f  the  S\ippt.—K{  l>  is  the  ditlerence  of  level 
between  the  wcir-<  rest  and  the  down-stre.im  surfaie, 

.  '/>  \    .r 


/•/,()»'   in'i:K    U  FIRS     HA/.IN). 


'■^i 


for    wiiiill  it  in  usually  ^iiliu-.i-nily  a.curati-  lu  subslitutf  ihr    aimpk-r 
expression 


-  1.05   +.1: 


These  formula-  .ur  only  true  (or  values  of  I)  between  certain  liiniis. 
If  //,  +  J)  is  t;i cater  than  about  3  i,  the  rise  is  moved  beyond  the  fooi  of 
the;  nappe,  and  the  formula?  in  the  precedinij  ca-sc  become  applicable. 
Again,  if  the  head  H,  is  not  suUicient  to  enable  ihe  nappe  to  push  bark 
the  rise,  the  down-stream  surface  level  must  be  sutliriciuly  hi,i;li  to  pre- 
vent the  admi>sioii  nf  air  Ix'low  the  nappe. 

The  drowned  nappe  preserves  its  characteristic  profile  even  when  the 
down-stream  surfaie  is  on  a  ievul  w.th  the  weir  crest,  Fi>;.  71  but  if 
the  iliHerence  of  level  between  tlie  up-  and  down-stream  surfaces  sti.i 
continues  to  diminisli.  a  point  is  reached  at  which  the  nappe  suddenly 
■md  with  an  iiiuliihitiui^  movement  as^ain  forms  part  of  the  surface.  This 
change,  which  is  veiy  apparent,  does  not  seem  to  have  much  influence 
on  the  coefficient  of  discharge. 


Kic..  71.  — Prowiicd  Nappe. 


i^ 


Kk;.  72.— .VdhcrinR  Nappe  (/A 
not  very  small). 


HE— T- .- 


.V 


■\ 


1 


Flo.  73.  — Adhering  Nappe,  spring- 
ing  clear  above  crest. 


Flu.  74.  —  Ailturing  Njppc  iff, 
smu'i 


On  certain  rare  o<-casions,  and  under  conditions  >;overncd  by  the 
thickness  of  the  wcr  an<l  by  the  construction  ol  the  iipi)cr  poition  carry- 
ing the  crest,  the  nappe  becomes  adherent.  Fi^s.  7i  to  yU.  the  sheet  of 
water  remaining  in  contact  with  the  wcir  face.    The  coclficu  nt  c'  is  then 


104 


ILOli-   OITR    HI  IRS    ifl.-l//,Vl. 


increased  anci  ni.iy  bccoino  as  large  as  1.3. .  Cdrrcspoiiding  Id  an  absokut; 
value  i.f  .55  or  .56. 

From  '.vhat  has  been  saic)   it  may  be  at  c  nee  inferred  that  the  dis- 
charge over  a  weir  is  largely    influenced    by   the    form    rif   the    nappe- 


-T^V- 


^rfrfrf^*^»yv^yy>.  ^'p 


FlO.  75.  —  Nappe  ailhcriiiK  "n  crest  onlj. 


Fk;.  7C. 


Taking,  for  example,  a  sharp-crested  weir  0,75  m.  higli.  it  was  found 
that  lor  a  liead  over  tlie  crest  of  0.2  in.  the  c<jeliicieiit  of  <'!scharj;e  i 
was 

.433  for  a  free  nappe. 

.46     "    a  depressed  nappe. 

.497    ■•    a  drowni  J  najipe. 

.554    "    an  adhering  nappe. 

litiim  Wi-trs. — These  weirs  are  foniied  of  squarecl  timbers  laid  one 
abiive  the  other  to  any  requiied  hrii;ht,  the  weir  faees  being  vertiral  and 
the  crest,  <ix  sill,  ha\ing  a  widtli  <■  e(iual  to  that  of  the  timbers. 

Free  A'lj//;'.  — The  nappe  may  either  spring  clear  from  the  up-stream 
edge,  when  the  case  becomes  that  of  a  sharp-ciesled  weir,  or  it  may 
remain  in  contact  with  the  sill  and  spring  clear  from  the  down-stream 
edge.  The  first  case  is  at  once  realized  if  //,  exceeds  ie,  and  may  occur 
for  any  value  <)f  h  between  ie  and  ij/,  the  change  being  produced  i)v 
any  such  extraneous  disturbing  cause  as  the  .idmission  of  air  or  tlie 
passage  of  a  floating  body,  etc. 

When  the  napi)e  remains  in  contact  with  the  sill. 


-K- ..«.','•). 


an  expiession  deprnding  essentially  on  the  value  of 


// 


For       =    .5  . 


=  1.0. 
=  1.5. 

=  I.o 


•79 

.88 

.^8  (  If  the  nappp  remains  in 
1.07  >      rontait  with  the  sill. 


ILOW   OyiiK    HIIKS     HA/.IS). 


Tlie  rati')     -   is  unity  fur  all  valurs  (,f      'ubuve   2,  .mil   if  tl-.c  iiamn- 
(  -  f  11 

spriiij^s  (  kar  Imm   llic   llp-^lrl^im   (•rl;^i-.  for  all  \  aliit--s  of      '  bctwi-iii   i '. 


Wall  sills  of  coii-iidcrabic  widt  li,  c.^.,  i  ot  2  m.,tliL-  aliovf  lorinula 
still  i;ivi's  iL'sulls  whuli  air  ap|)ri)xiiiiatelv  correct.  The  latio  '  mav 
(litiiiiii^h  to  .1  few  tenths  or  even  less  than  .  15.  With  a  J-ni.  tl.it  (  rested 
\vi  ir    experiment  y.ivc   for  a  head  of  .45  ni..       -=  .755.  the  eurrespoiuhiii; 

absolute  value  of  <'  Ijevnj^  .337.      The  formula  i;ives'    —  .742,  the  coire- 

spondinu  value  of  <'  bein<;  .331. 

The  roiiiiiiiiij^  o(  tlu;  up-stream  edge  t)f  tlie  sill  lias  a  very  sensible 
infliieiH-e  upon  the  How,  and  the  effect  of  a  radius  of  only  1  or  2  cm.,  as 
usually  results  from  ordinary  wear.  n)ust  by  no  means  t)e  disregarded 
in  gautiing  the  discliarH;e.  Ftelcy  and  Stearns  observed  that  the  effect 
of  a  small  radius  K,  not  exceeding;  J  in.,  or  0.012  m..  was  to  increase 
the    head    by    .7/1'.    and     therefore    the    CKlTicient    c'    in    the    ra;io    of 

//a'  to  (//.J  +   .7/1')'.  or  approximately   1  to   1   +    ..    ■       This  .ipj/o  .xima- 

lioii  IS  not  suflicienlly  accurately  for  sensibly  greater  radu.  With  iwr) 
weirs,  tiie  one  .8  m.  anfl  the  other  2  m.  wi<le.  tli'-  up-stie.im  edges  being 
rounded  to  a  radius  of  .10  in.,  the  discharge  was  increased  14  percent  \i\ 
the  lirst  and  12  per  cent  in  the  second  case.  Wilii  the  j-in.  weii  the 
roetficient  <'  for  the  greatest  head  used  in  the  experiments  w.is  found  to 
be  .373,  which  is  very  nearly  the  same  as  the  value  theoretically  deduced 
on  the  assumption  tliat  the  flow  over  the  weir  is  in  fluid  filaineiits  par- 
allel to  the  sill.  This  condition  is  only  imperfectly  real. zeil  in  practice 
as  the  surface  of  the  iiapiie  invariably  has  an  undiilatory  movc^nient. 

/>f//cv.i,(/  luiil  Dro-vitiil  .\apf>ts.  With  a  sharpcrested  weir  the  co- 
cllicienl  for  a  depressed  nappe  is  alw.iys  greater  than  ih.it  for  a  free 
nappe.  With  a  beam  weir,  such  as  that  now  under  consideration,  the 
coedicierits  dilfer  only  slightly,  thai  for  the  depressed  weir  being  at  first 
a  little  less,  then  about  the  same,  and  tinally  a  little  greater  than  the 
<:oelRcient  for  the  free  nappe.  When  the  ri.ippe  is  lirinuned.  the  influence 
of  contact  with  the  sill  is  complicated  by  li.e  fact  that  it  is  impossible  to 
«lttlne  exactly  the  point   at  whu  h  the  n,ippc  is  freed  from  ihe  sill,  and 

this  separation  no  longer  corresponils  to  a  certain  constant  value  of       '. 

e 
It  may  again  occur  cither  before  or  after  the  establishment  of  the  ihinvnrtt 
lundilioii.     Two  cases  may  be  distinguished.      If   1  fihe  height  of  v  eir) 
.>  5^.  the  separation  lakes  place  in  advance  of  the  drowned  slate,  and  in 
this  intermediate  condition  the  napjic  does  not  differ  from  that  which 


io6 


//();<-  oii-R  n'/:ms  ^ha/.in) 


flows  over  a  sharp  tixslcd  weir.  Il  i  ■  y\  tlu'  nappe  is  not  freed  from 
tlie  sill  befoie  il  assumes  the  dnnviieil  loriii,  and  at  the  nioninit  of  the 
change  is  very  unstable. 

Si>  lotii,'  as  the  contact  .vith  tlie  sill  coiitiiuies.  its  infhunie  predoini- 
ii.ites,  and  the  formula 

\  '   / 

is  f.iirly  api)li(al)le  to  llie  drowned  nappe. 

Hut  when  the  nappe  has  left  the  sdl,  the  phenomenon  becomes  more 
and  more  nearly  the  Mime  as  for  a  sharp-crested  weir,  and  the  lornuihi 
now  applicable  is 

.■=..(.S;8  +  ..38;j-. 

yive  the  same  value  lor  <'  for  a  certain  limiting 


These  two  foimul.i 
■alue  of  //..  t;iven  by 


I   + 


♦■  '•)■ 


//■-. 
holds  when  the  heads  are  less  than  II,'.  but  the  co- 


The  first  formula 

cllicients  are  ;i  little  too  small  althounl  the  eriors  are  never  more  than 
3or4pcr<ent.  If  the  heads  are  jjreatcr  than  //•/.  the  second  formula 
is  to  be  used,  btil  the  results  art;  again  too  small  and  the  error  m  this 
case  ma\  be  as  mui  h  as  S  per  leut  at  tin-  moment  when  the  nappe  is 
sep.irated  friiiii  tin-  >iii.  The  error  then  ia|)idly  diminishes  as  the  head 
incii'ases. 

lli,i,-,i,yl,ii  U'l/rs  -.rit/i  SA'/>i/ic  /'•'"S.  —  ln  these  the  coetricient  ^. 
depending  upon  the  he;id  (//,).  the  width  (<>)  of  crest,  and  the  degree  of 
face-slope,  is  iiiiw  ixlrcmely  variable  ami  eacii  case  must  be  sub- 
jected to  a  special  invcstigatitm.  The  face-slope  on  the  up-stream  si<ie 
lias  the  effect  of  diminishing  the  contraction  and  therefore  increasing 
ihe  discharge.  The  downstieam  face-slope,  on  the  other  h.ind,  pro- 
iluces  an  effect  similar  to  the  wideniiii;  of  the  crest  and  dimuiishes  the 
(lisrharge.  The  rounding  of  the  up-stream  edge  of  the  crest  consider- 
ablv  diminishes  llu'  cimtraction  and  may  inc.  ..se  c'  by  to  or  15  per  cent. 
Finallv.  <'  is  verv  largely  increased  for  weirs  with  1  ompleiely  curved  pro- 
tiles. 

Ma/in  bus  prep.ired  Tallies  (ompn-ing  .1  sntficient  number  of  partic- 
ul.ir  cases  wliich  may  serve  as  . I  guide  in  praitio-,  it  is  impr.irti<able 
In  establish  a  general  fornuila  which  will  take  '  ito  .iccoiint  all  the  vari- 
able elements  referred  to. 

J)r,<-nii,,i  l(V//>  n///'  S/iar/>  t>.>7.>.  When  the  water  on  the  down- 
Stream  side  dues  not  stand  mutll  above  the  crest  of  the  wcir.  ii.i/in  gives 
the  somewliat  1  ■  nipii<  ated  forniui.i 

I  /'       I        ,.       I  /'       1   //'\'  I    ■«■ 
-   =  i.o^  +  -  -^  .ooi>  +  +   ,  I    ,       1'  //"■ 


1 


1 1  OH'  Ol'F.K    Hl.lRS  {H.I/.IN> 


107 


III  the  niai'inty 


ix'S.  liinvcvcr.  llif  follo'.vinj;  siinp'u-r  formula  is 


=  li.u.S  +  .iX— 


3  //i  -  J' 
\       II'      ' 


These  two  formula'.  estahlishe<l  so  as  to  re|)rcseiit  as  accurately  as 
piissiblc  tile  particular  expiTimcnts  by  winch  tliey  have  been  deduced, 
mav  be  replaced  by 

//,'\  •  117-  i> 


■.vliich  uili  iiwc  resuils  dilTcriiii,'  Iroin  those  obtaiiieij  with  the  other  for- 

iiiuUe   bv  not   more   than   .ibout  1  i>r  J  per  cent,  unless         and        "   are 

veiy  small,  when  the  ditl'ereiue  may  be  as  much  as  4  or  5  pir  cent,  but 
in  the  1. liter  case  the  determinaiion  ol  ('  Is  always  somewl.at  uncertain. 
The  etteet  of  drowiiini;  is  not  the  same  for  wide-crested  weirs.  The 
flow  on  the  up-stream  side  is  not  alTected  bj  the  depth  of  the  water  on 
trie  down-stream  side  until  the  downstream  surface  rises  considerably 
ab^ive  the  weir  crest,  aiui  the  ellect  diminishes  as  the  width  of  the  crest 
increases.  In  the  case  "i  a  sharp-crested  weir  the  influence  u|)oii  the 
ip-slrc.im  flow  is  felt  hejorf  the  diiwii--treani  surface  has  readied  the 
level  of  the  crest.  As  the  width  of  a  wide-crested  weir  increases  it  loses 
its  weir  characteristics  and  approximates  more  and  more  closelv  to  an 
open  channel  with  horizontal  bed. 

/"///('(•//.•>.«  of  .\'<ifi/>e  on  U'l-i)  t>(>/.  —  Let  /  —  thickness  of  nappe. 

For  a  >//cr;/-(->r.>/<'</ weir  and  />>>•  nappe  -.     \aries  from  .85  to  .86. 

/ 1 1 


For    a    s/iarp-i  > estfii  weir 


bein;;   .S   wh?n        ^   .4.  .S55   when 

.1  .1 


:ii   n.iiipe   ,,   increases  with 

II  I 


I.  anil  .87  wlieji 


■As  the 


ilown-stream  level  rises         incre.iscs.  e.\i  eiiliiii;     t)    for    tin     undnl.iting 

C'     •iition,  and    necessarily   tends    to   unitv  as    liie    ditference   of   level 

liclween  the  crest  and  the  down-stre.im  surface  is  i^reatly  dmiinished. 

,      .  .  ,     ,  I  ,  ,,       .  ,  /' 

In  beam  wcirs  with  fret  nappes 


.9  for  sui.dl  values  of  -,  anri  de- 


cioases  as  the   he. id    increases   until    the  ratio  becomes  .(5,5.  when   tiie 
nappe  is  on  the  point  of  separating  from  the  sill. 

In  heaiii  weirs  with  («r(i.t'//(</ n.tppes  the  van.ition  of  is  somewha; 

CotnpUc.ited.      The   ratio  diminishes  until   a   mininuim   i>   reached,  and 


Im 


'°^  hERXOLlUrs    Im.OKH.M. 

tl.-M  iMc,c.,s,.s  a,„l  .ipproxnn.acs  tu  value<  wind,   ,r,-  ,1„.  « 

«-as.-  ..f  sliari,-rr,st<-<l  w.-,rs.  '"'"^  "'^  '"  '''* 

III  Tiv;/,--,;,.!/^,/ weirs  with    i/„/./.r.'    f,  '     ■ 

till-  up-Mrca.n  face  hv  a  .urvi.l   Mirfa-e     ''    ,„•„■  I      i 

<l.t.T,„i„^„i,„,   of  iho  nappe  thirkno.  ,.  i„ 
si'<!i  .aMniiM  I,  l.ss.Kviir.it.- 

28.  Bernouilh-s  Theorem.-A  s.mplr  pp  ,of 
<•!  this  tiic'inm  is  as  fallows: 

("nsKieran  iruit-finitelv  small  rl,-,,,,.,,!  <,f 
a  Mua»,-h„e.  of  Icnoth  ,A  a.,.1  sfctional 
ana  ,?. 

I..t/,  /  +  ,//hrtli<-  itilcMsiticsof  pressure 
•  it  tliL-  eiuis. 
•••  hv  the  sp,.c,ti<-  weight  .,f  the  fluid. 
•    ■'      ■•    allele  hetwecn  the  .lireriioti 
-f  niotioM   of  the  .-leiMeMt  and   the 
\ertical. 
■f~    Uv  the  verticil    i.r,.iectioii   of    ,/. 
so  that  ,/c  :^  ,A.eos  ,1. 
!<es,,Ivin^;  i„  the  .hreetiun  of  „„„,„„, 

t  .  ./>  r,,.  ,,     ,  axeloratiiig  f,,ree 

-^  iilas.s    .    acceleration 


^      a  .  ,A 


—      </ 


.V 


.// 


./■<: 


'//> 


A' 


Intenratriu.   :    ...     /'">    .    ■  '   _ 

•       •"        ^"^  ~  ■'   ''"'"*'••    '"*   ""lie   f..r    any  lluid.      ff 
lt>e  fluKl  ,<  w.ucr,  :..  is  constant,  and  then  r    ,    ^  +  ■'!  ^  ., 


const. 


F.X.-IMriMs. 


lor) 


EXAMF-'LES. 

(N.  H.    In  th.-  follow,,,;-  ox..„,,.l.-s,^  -.  y.  ,„,i,ss  otherwise  speri,„v|., 

..     /tons  of  water   fai:  //  f.ct  per  ,n,n„te  a.ul    ,re  employed  to  Vnn 

tur„nes  wh,.l,  transfor,,,  ,„to  usef.l  work  on.-  half  „f  the  t!,tal  ener^J 

«f  the  water.      U  hat  is  the  H.R  of  the  turhinr,  •>  /,f^^ 

■  hi ..     -     . 

2.  A  f.rb.ne  transforms  ,nto  ,.;.  IM'.  ■,;  um-IuI  work  the  ei^Jr-.y 
Of  the  water  fall,„.  .J  ..-et  fromaTh„m.o„  V-,„,t.  h,n  wh,eh  the  w atT-r 
Stands  at  a  constant  level  2\  ft.  above  the  bottom  of  the  not,  h  n  the 
coefficient  of  discharge  ,s  .6  what  ,s  the  etiicency  of  the  turb,ne.= 

3.  A   fall  ul    ,oIt     supplies  to  a  turb.ne   ,.  ru.   ft.  of  water'per'L 
The  turb,ne  uses  only  «  ft.  of  the  fall,  and   the  water  leaves  the     urb,'  ^ 
w,th  a  vel..c.ty  of  8  ft.    per  se--.      If  500  lbs.-,t.   ,.re  lost  ,'>',," 
sistan.e.  etc..  I.n.l  the  ellicien.y  of  the  turbme.  'MUional  n  - 

4.  10,000  50-volt  incandescent  and    -;o  j-o  u  .ti    .r  •  1        "   "' '   ^^' 
.npphed  with  power  fron.  a  wa.erfa,,   h:r.;:;°  ri^ ,J;';rd'::;;'f:"^ 
20   „„U.s  d.stant^      Losses  between   l.unp.  and  converting    .,pp.„a,' s    a 

:r;;:;L'::r::"';::; ,;;::; •■'" -"--;5«,..  ,<:;;:;;;j"r: 

.      A,      .        ,  ■'"'•    '•'^'^^•'^>.iO(u.  ft.  r)er  honr 

;;:,;,";/;,„';■;.""■""•  ■■""  ■'-'■—  ■■■  .;"-'-.i':';;  :',;r,;"; 

f     \         I  ■     ■      ,        ,  •!'!>.  504.6  lbs.  per  s(i    ft 

6.      n  a  fr.cfonless  horizontal  pipe  d,srha,«,M«  ,ocu.  ft.  of  water  per 
cond.  the  di..n,eter  gradually  changes  fro,,,  4  in.  at  a  point  ./t  .T 
.1  a  point  A.      I  he  pressure  at  the  po,„t  /.'  is   ,00  lbs.  per  sou  .re   „ 
hnd  t.ie  pressure  at  the  point  .,.  . ,,,  ,,,J,„^   ^^   '  -"• 

/.A  ,.in.  hor,/ontal   p,pe  1,  :;rarlualiv  redu,  ,-d   m   diameter  to   4  in 
and   then  gradually  expanded  a«am  f,    u.   ,n..utl,     when 
the    atmospiier.      Determine    the    ma.„„.„    .,,;X   ,  ::::Z.X 
can   be  forced   through   the  pipe   U,    wh,„   tne.li  .meter     ^ih 
.s  i  in.,  (fi)  when  the  diameter  is  |  i„     Also  .Jetern,     • ,.  '""""' 

velocme.  at  the   throat  and  thj  totaU;;.;;'    ^     fnZ:":;"'^ 
however,  is  very  considerat)lei.  ^  ir.ction,  which. 

.■l»s.^.,>  ..-^cii.  f<,  pcrnni,.,  46-  ft.  per.  sec 
(*'  .-3-VCu,ft.  per  mm..  46.66  ft.  per  sec. 


lO 


I  X.-IMri.HS. 


8.  A  short  lii.ri/omul  pjpe  .//>'L  coniifctin^  two  nst-rvoirs  Rratliially 
ojntraits  in  diatiieicr  fn.m  i  m.  at  .•/  to  J  in  m  A.md  then  enlarges  t<j 
I  in.  again  at  C.  If  tla-  height  of  tlio  water  in  tlic  rt-crvoir  uvt-r  C  be 
12  ins,,  ilctcrinine  the  inaximuni  flow  ihrou^li  tlie  pipe  and  sketi  li  the 
curve  ol  prosures.  Also  obtain  an  equation  fur  lliis  (.iiive,  assutniriji 
tlie  rates  of  contraction  and  expansion  of  tlie  pipe  t.i  be  equal  and 
""'f'"'"'-  ./'/>.  4  <  u    It.  per  nun. 

9.  In  a  iliverguij;  niontiipiece  the  rjiametei  of  the  tliro.it  is  .6  in., 
and  the  head  of  water  over  the  .i.xis  is  30  ft.  Wli.it  is  tlie  dischar<;e  in 
j;allons  per  minute  when  the  v.icuiiui   ,,t  the  thro.it  is  1.S.3  ins.  of  iiier- 

10.  In  a  stream  with  still  water  240  ft.  above  d.ituin  ami  flouinj; 
without  fricti.in,  the  veloeity  at  a  point  15  ft.  above  d.itum  is  24  It.  per 
seciuid.      Wiuit  is  the  pressure  .it  this  point? 

.Ins.  10S.75  lbs.  per  sq.  in. 

11.  .\  funnel-'haped    moullipiece    leads  from  a  reservoir  into  a  6-in 
fiictionless  pipe,  so  that  tlierc  is  no  contraction.     Tiic  water  flows  with 
a  velocity  of  24  It.   per  .second.     Kind  the  pressure  at    a    point  in  the 
pijic  10  fl.  be:nw  the  sutf.ice  of  the  water  in  the  reservoir. 

.///>.  I  5,43  lbs.  per  Pip  in. 

l^   .\  3-in.  pipe  gradually  expands  to  a  bell-mouth  ;   if  the  total  head 

//,  be  40  It.,   lind  the  i;reate,st  diameter  of  the   mouth   at  which   .     will 

run  full   when  open   to  the  atmosphere.     Comp.iic  tlu'  ilischariic  from 

thispi|ie  with  the  ilischaii;e  when  the  iiipe  is  not  exiianded  at  llie  moulli. 

.///-.  4..S    in.  :  discliarse  is  149.070  en.  ft.  per  mimte  witli  bell- 

nioiiili  ,m,.   .;.345  cu.  It.  per  minute  without  bell-  li. 

13.  The  pressure  in  a  12-in.  pipe  at  A  is  50  lbs.  jier.  sq.  in.  :  the  piiie 
then  enlarges  to  a  15-in.  pipe  at /.'.  the  riM'  from  .■/  to  A' liein);  3  ft., 
the  di.-(harf;e  is  1100  cu.  ft.  per  minute.  Find  the  piessureat  /.' ;  aKo 
liiid  the  piessure  at  .1  point  C',  the  rise  from  />'  to  C  being  G  ft. 

.■Ins.  •■I42>  lbs.  per  scp  ft.;  6/67J  lbs.  per  sq.  ft. 

14.  One  cubic  foot  of  water  per  second  flows  steadily  through  a 
frictionless  (lipe.  At  a  point  ./,  loo  ft.  aliove  datum,  the  sectional  .i.e. i 
of  the  pipe  is  .125  sq.  ft,,  and  the  pressure  is  25W  lbs.  per  sq.  It.  Find 
the  total  energy.  ,\t  a  point  />' in  the  datum-line  the  ppsMire  is  1250 
lbs.  jier  sq.  ft.  and  the  sectional  area  is  .0625  sq.  ft.  1  iiid  the  ioss  of 
energy  tietween  ./  and  /.'.  Find  the  "  loss  in  shock  ■  if  the  sectional 
area  at  A'  abruptlv  changes  (,;)  from  .u;  to  .0625  scp  ft,  ;  1/"  from  ,o^2; 
lo  .125  sq.  It. 

.tns.  141  It. -lbs.  ;    117  ft,-lbs.  :  79  ft. -lbs.  per  cu.  ft,  ;  (.jj  fl.-lbs. 
per  cu.  fl. 

15.  In  a  frictionless  pipe  the  diameter  graduallv  changes  from  '  in. 
at  a  point  ./  20  ft.  above  datum  to  3  in.  at  />■  [5  ft.  above  datum.  The 
pressure  at  A  is  20  lbs,  per  sq.  in.  :  find  the  pressure  at  /!.  the  deli\ery 
of  the  pipe  being  23  cu.  ft.  per  sec.  ,l„s.  2,23  lbs.  per  sq.  in. 


/■x.-iMr//  .s. 


II I 


I*..   A    iiDi  izuiu.il    frictiunli-'^s 


pipe   i^r.idaally   CMiur.icls   t,)   a   thi...it 


of   -l!i  i.f  tho   area  ami  tlirn 


^'radnally  (■ri!ars,'cs  ..naiii    to  a   pine   ni  the 

same  size.     ]f  /is  the  velocity  i.I  ll.-w  ni  the  |iipe,  lind  tlie  rcdiutiMii  of 
pressure  at  the  tliroat.  IIT' 

./'.•>.        ^       (//'  —  n. 

I-.  Tlie  pressure  in  a  3j-in.  liorizontal  frictior.'ess  pipe  is  6;;  Ihs.  per 
M|.  m.  above  tliat  of  the  atniospl-ere.  The  pipe  is  Kra(hially  re(.iice<l  to 
a  throat  ol  on,-  fifth  of  the  area  and  discharijes  into  the 'atmosphere. 
Find  the  velo<  ity  of  emux  and  the  amount  of  ilie  discharije  in  {gallons 
!"•'■  "-'""tL-.  A„s.   97.9S  ft.  per  sec.  ;  491.177  j-aN. 

18.  A  frictionless  play-pipe  ^radii.illy  e.xpands  from  a  diam.  of  i  in.  at 
the  I)ase  to  a  diain.  of  3  in.  at  the  mouth.  There  is  a  discharge  of  33  cii. 
ft.  per  mill,  under  a  head  of  183  feet,  l-ind  the  .-oetricient  of  discharge! 
the  force  reipiired  to  hoM  the  nozzle,  and  the  tot.d  11. 1',  developed. 

./'/f.   .9:!",;   loSii  Ihs.;   11.56  H. I'. 

19.  lind  the  disdiarge  in  <  iibic  feet  per  minute  under  a  head  of  2  ft. 
ihrou^rh  a  liorizontal  frictionless  pipe  which  -radually  diminishes  from" 
a  diam.  of  \  in.  to  a  thn.at  of  \  in.  diam..  at  wiiich  the  pr.  head  =  (>  iiis.. 
and  then  ^.-radually  enlarges  to  a  pipe  of  .same  diameter  as  before. 

■  Uis.  .:i>i~. 

::o.  I'ind  the  hea.l  required  to  !,'ive  l  cu.  ft.  of  water  per  .seonid 
throui;h  an  orilice  of  2  sipiaie  inches  area.  Inc  coeilicient  of  dischap'c 
'"'i"K-"-5-     (.f  =  3::.)  .l„s.  307.36  ft.'' 

:;i.  The  area  of  an  onlice  in  a  thin  plate  was  y,.j  s(juare  centimetres. 
the  discharge  under  a  head  of  3.396  metres  w..s  f.nind  to  be.  .01825  cuI'k* 
i;et..  |ier  se.-.)nd.  and  the  velocity  of  flow  at  the  contracted  section,  as 
determined  bv  measurements  of  the  axis  of  liic  jet,  was  7.98  metres  per 
.secon.i.  Find  tlie  coelfK  ients  of  velocitv.  contractieii.  dischar-e  and 
lesistance.     (<=9.8i.)  ■  I 'is.  .gyy  :  .(,32:  .6,6 -^.046. 

22.  The  piston  of  a  1  2-in.  cylinder  containint;  .salt-water  is  pressed 
down  under  a  force  of  3000  lbs.  Find  the  velocitv  of  .■tllnx  and  tin- 
volume  of  discharge  at  th,-  end  of  the  cvbnder  through  a  well-nai.ule.l 
l-in.  orihce.      A\>n  lind  the  power  exerted,  ,,.  beiiij^  .977  and  c  =  .5341. 

,/«.(.  60.373  ft.  per  sec.  .  .176  cu.  ft,  per  sec. .   1.166  II  I'. 

23.  In  the  condenser  of  a  marine  entjine  tiiere  is  a  back  pressure  of 
26S  in.  of  mercury;  the  injection  orifices  are  6  ft.  below  the  sea-l.'vel. 
With  what  velocity  will  the  injection-w.itir  enter  i  he  condenser  .^  1  Net;- 
le<t  resistance  atul  lake  ^  r-  32.2.)  _,,„_   ,53  f,    ^.^^  sec' 

24.  \V.iter  in  the  f.ed-pipe  of  a  steam-engine  stands  12  ft.  above  the 
suiface  of  the  water  in  the  boiler;  the  pressure  per  sq.  in.  of  the  steam  is 
20  lbs.,  of  the  atmosphere  .5  lbs.  Find  the  velocitv  with  which  the 
water  em.rs  the  boiler.  ,•„  bcie,   .97,  yOn,   5.376  ft.  per  sec. 

=  5    ''lii^  injection  orilicc     .'  a  jet  condenser  is  5  ft.  below  sea  level 


1 1: 


I  x.iMPJ  i:s. 


and  x.Kumn        -  in.  -I    ni.Tiuiy.      l-ii"l  \<'l.i(iiy  "f  wmUt  cMti-rmn  coii- 
(loiisiM.  supposing  tlirc-  fourths  ..t  the  lii-.i<l  U»\.  Iiy  fii.tional  ri.-sislaiue. 

.  hit.     J3..S')  ft.   piT   SCI-. 

.").  Tlic  ill  fiipin  .111  oiifuf  of  .008  s<i.  ft.  area  m  llic  side  ol  a  l.uik 
and  iindi-r  a  head  of  r6  ft.  issues  huiizonlally  and  falls  I  II.  \erti<  ally  m 
a  hori/untal  ran-e  of  7.6X  feet.  The  delivery  is  60  -allons  per  minute. 
Find  the  eoefli(  ients  ol  velo,  iiy.  disi  har>;e,  ronlra.  tion,  and  iesi>tan<e. 


.  Iiii.   .'/> ;   .h2 


;  .6;  ;   .0S5. 


27.  Tin;  jet  from  a  (  in  ular  -h.irp-e.ln.'  oritiee  J  m.  in  diam.  iindei  a 
liead'  of  I.S  ft.,  stiike>  a  point  at  a  distanie  from  the  oiiliee  of  5  It. 
measured  li(.ri/orit,dlv  and  4.'.(>5  ft.  measured  verlie.dlv.  The  dis- 
<liari;e  is  yS.^S;  i;allon>  in  51.0.21S  mm  ouils.  I'ind  the  ro.lh.  leni^  ot 
flischar^e,  velot  il\,  luntrai  tion.  .md  resist. mee. 

./«>.  .(Kxi(>;  .>m;;  -''35:  •"'/'■ 

251.  .\  sluice  3  fl.  xpiaie  and  witli  a  liead  of  13  ft.  over  llieientre  has. 
from  tlie  ihiikness  of  the  frame,  the  .ontnietion  suiipressed  on  all  side^ 
when  fully  open;  when  partially  <iiK-n.  the  (ontr.uMion  .-.xi^ts  on  the 
upper  edj-c.  i.e.,  aij.iinst  the  hollom  of  the  n,ilc,  wliiih  is  f..ime(l  of  a 
thin  sheet  of  nielal.  Find  the  dist  liarj-e  in  <  uhie  feet  wiieii  opnied  1  ft., 
2  fl.,  ami  also  wlieii  full\    ■pen.  -/'".   57.22;   llv.V'^;   I7.V5"' 

29.-  A  vessel  eontaminn  water  is  pl.ued  on  stales  ami  weinhefl.  How 
will  tiie  weijih:   he  atTecteil  l>y  opeiiiiiK  a  small  onfuu  ill  Ihc  holloiu  ul 

the  vessel  ? 

30.  Water  is  supplied  hy  a  m  oop  to  a  li>comolive  tender  at  7  feet 
ahove  trout-h.  Find  lowest  speed  of  tram  at  whur  the  opeiMtion  is 
,,„sj,i,le.  •'"'•   '4-44  miles  per  liour. 

.\Uo  find  the  velocity  of  d.  livery  when  train  travels  at  40  miles  per 
hour,  assuming  half  the  head  lost  hy  frielional  resistame.     (.,.  =  1.) 

./mv.   35.68  ft.  j)er  second. 

31.  The  head  in  ,1  prismatic  vessel  .a  the  instant  of  opening  an  onlue 
was  (>  f'.  and  at  elo^ini;  it  had  deeieased  to  5  ft.  Determine  the  mean 
eonsiant  head  /;  at  wlm  ii.  in  the  same  lime,  the  oritke  would  disdiarj-e 
the  s.ime  volume  of  water.  ■''"•    5  4***!  ''• 

32.  A  eylindrieal  vessel  5.747  in.  in  diameter  has  an  orifice  of  .2  in. 
tliam.  at  the  li..ltom  ;  the  surface  sinks  from  16  in.  to  1  2  in.  in  53  secmds. 
Find  the  coelFu  lent  of  discharj^c.  «-<»"■    b- 

33.  .\  prismatic  basin  with  a  horizontal  sectional  area  of  9  sq.  fl.  lias 
an  orilice  i.f  <;  s(|.  in.  at  the  bottom  ,  it  is  lilleil  to  a  depth  of  r.  ft.  above 
the  centre  <•(  the  orifice.  Find  the  time  requireil  for  the  surf.u  e  In  sink 
3  ft..  3J  (t.,  5  ft.  «■'"*•  258.9  sec,  500.1O  sec.  ;  834.8  sec. 

34.  The  water  in  a  cylindrical  cistern  of  144  sq,  in,  sectional  area  is 
16  ft.  deep.  I'lMm  opening  an  orifice  of  1  sq.  in.  in  the  bottom  ihe 
water  fell  7  ft,  in  1  miniiie.  Find  ihr  coefFicient  of  riisi  har^e.  The  co- 
riricienl  of  contraction  being  .625.  find  the  coeHicicnts  of  velocity  and 
rpsisiance.  ^"'    -^  ■    V6  ;  o-8S- 


FX-IMPtMS. 


35.  H'nv  lonn  will  it  t.ikc  to  I'lil  a  parabuloidal  vessel  up  In  tlie  level 
of  the  nutsidp  surface  tlmiugli  a  hole  in  the  bottom  2  feet  under  water? 
(j^  =:  3:  aiifi  <•  =  /i:;;. ) 

/liii.  .,  /•'  bein''  the  parameter  of  the  par.ib.il.i  and  .•/  the 

.05     ; 

sectional  ari.i  of  the  irilire. 

36.  H<i\v  lori^j  will  it  take  to  till  a  spherical  vessel  of  r.iijinsr  up  to  tlie 
level  of  tlie  outside  surface  throujih  a  ImIc  of  area  ./  at  the  lowest  point 
an<)  2  fl.  under  vvatei.  i  beini{/i25  ? 


^;/i 


v/ 


•54' 


6.53)- 


37.  A  ;(>><,'all<)n  tank  is  loo  feet  above  ground  and  is  tilled  by  ii  ij- 
ineii  pipe  eoniieeteil  willl  an  aicninul  ilor  havinn  a  ;-ft.  <\lr.  piston 
loaded  with  50  tons,  ll  tlieiiuMii  lilt 'i(  tlie  pslon  i-  10  fi.  and  if  ,•„  o( 
the  head  is  lost  in  Iim  tional  resistance,  how  Iniiv;  wi,!  it  t  ke  to  till  the 
lank?  -'">.  1449  Sfcs. 

3S.  A  bucket  of  Water  in  a  balance  dischar^i's  4  liis.,  of  water  per 
rainiile  tlironj^h  an  oiiiico  in  its  base  at  43°  to  the  vertical,  and  is  kept 
coii>lantlv  full  by  a  vertical  stream  which  issues  from  an  orifice  8  ft. 
abo\e  the  surface  with  a  vidocity  of  30  ft.  per  sec.  Show  that  the 
bill  ket  must  be  counterpoised  by  about  .o6fi  lb    in  Te  than  its  weight. 

39.  The  water  in  a  vessel  <}  ft.  in  heijjhl  and  .;  ft.  in  diameter  is  8  ft. 
deep.  In  wh.it  time  would  one  hall  of  the  w.iter  How  away  llirou|.;li  an 
oritire  III  the  boito'n  1  inch  in  di.imeter  .*  If  the  ori'icc  is  closed  and 
the  vessel  is  iii.ide  to  lotati"  about  its  .ixis  a!  the  rate  of  -f(,*,  revolutions 
per  niinutr.  to  what  lieiylit  will  the  w.iter  risp  on  lilt*  vessel's  surface? 
If  the  orifice  is  "peiied.  find  velocity  of  ellliLt  when  the  surface  at  the 
axis  IS  \  fl.  above  the  orifice.  .Also  lind  the  dillerein  e  of  prcssurc-he;iH 
in  .1  hoii/ontui  plane  f>  inches  from  the  axis. 

Alls.  ii^).77  sees  ,  to  the  lop  ;   16  ft.  per.  sec.;  3  ins. 

40.  \  cylindrical  vessel,  10  It.  hi)>h  and  1  ft    in  di.iineter.  is  li.ilf  full 
of  water.     Find  ihe  number  of  revolutions  per  minute  svhich  the  vcsstd 
iiiiiNt  make  so  that  the  water  inav  iust  reach  tlie  lop,  the  axis  of  ri"    ', 
lion   beiiiK  (I)  coincident    with   the  axis  of   ihe  vcisel.t:)  ,1  ^^^•n   < 

line  •)f  the  vessei.  ./«(,( 1 )  4K3  ;  U>  .'., 

41.  A  vessel  full  <'f  water  weighs  3;^  lbs.  and  is  laised  xerticily  by 
nie.iiis  of  a  wei^iit  of  430  lbs.  l-'ind  the  velocity  of  etiliix  through  an 
orilice  in  the  bottom,  the  lii-ad  beinn  4  It.  aml^'  =  3^.2. 

Am.  17.01  ft.  per  sec. 
4.:.  .\   vessel   full    of    w.iter  makes  loo   revols.  per   iiiin.     Find  the 
veliH'ity  of  elHux  tliroui;li  ,111  orifice  3  ft.  b«dow  ihe  surface  of  ihc  water 
at  the  i:cntrc.  the  diam.  of  the  vessel  bcin^  3  ft.  and  <*=  I. 

Ans.  33.4  ft.  |)er  ler. 
What  will  be  the  velocity  if  the  vessel  is  at  rest  ? 

//'M    1 1.3  ft.  per  sec. 


I 


114 


r.XAMriiiS. 


f"  "  -,  43    Show  that  when    ilie  water  flowinp;  over  has  a 

■-,  '^ deptli  Rreator  tlian  .3874  ft-  '^  '■*  c.iniL-.l  .  ..iiipleicly  ov.i 
"T   ■  ihL-    loiiKituiiinal    ..[.cniiiu.    .S3    It.    ill    width.     At   what 

1  depth  (in,s  ,///ihc  water  11  .w  in.-  W«>.  .m  ft. 

Fig    77-  44    A    Mpian-    b»\    -    It.  in  lt;ii;;tli  ami   ;  (t.  acior--^  a 

diasiot.al  is  placMl  with  a  .liaK-nal  veti  .  al  an.l  lill.-.l  w.th  water.  IL.w 
lotiK  w.ll  it  take  f..r  the  wh-.lc  uf  the  w.il.r  to  llow  nui  throuuu  a  h.  ur 
at  the  h..tt..m  of  .0::  sq.  ft.  area  .-  w  ^  .0^5.)  .('".  97.5^  ■^'-^^ 

45  \  pv.aini.l  2  It.  hiuli.  .m  a  -quare  l.ase.  is  mverto-l  aiul  hW.d 
with  water  '  rmd  the  tnne  u.  whiM.  the  water  will  all  run  nut  tlinMiuh 
.,  h..le  uf   ...2  s.,.  ft.  at   the  ape.x.      A    >:.l.-  ul   the   hase  is  ■  It.  in  ler^;tl.. 

46  "Fi.ul  the  diseh...x.--  under  a  he.td  ..f  2;  it.  tl:r..u;;h  a  thin-lippe.l 
square  untke  of  .  sq.  m.  s..  lior.al  are,,.  (,..  when  it  lia.^  a  hor.ler  on  oue 
side   (/'t  when  it  has  a  border  on  two  si<ie>. 

.Ills.  (,n  .  -or*'  eii.  It.  |ier  see.;  (/)  r-,:^'  •  ".  ft.  per  see. 

4-  \  vessel  in  the  form  of  a  i.aral.oloid  ol  revohilioii  has  a  .l.'lith  ol 
,6  in.' and  a  .liam.  of  M  ,n.  at  the  top.  At  the  l.ottom  .s  an  or.t.ee  u! 
,  sq  Ml  serti.aial  an...  H  water  llow-  into  the  \es-el  ..t  the  rate  o  2,j 
,  ub,e  feet  per  rninule,  to  what  level  will  the  water  ullimat.lv  rise.  low 
lonnwillitta.eloiiseci)  ..  in.,  i/o  .  l.o  in.,  u  I  I  l.-vy  in  .  „/ (  I  -  m.  ahove 
Ihe  onlue'  If  the  sui-ply  is  now  slopped,  how  h.tiK  ..  wi;l  it  lake  to 
emiity  the  vessel  ? 

Aii^.  \l  inches;  ci)  49-17  s'"C. :  C')   '24-2  see.;    .     J.)J  se<  .  ;  1,1 
an  inliniie  h-iiHlh  ol  lime;  u\  11.3  sec. 

48  If  the  vessel  m  l-xample  47  is  ;•  sphere  1  ft.  in  .h.nneter,  to  what 
heinht  will  thewater  rise  '  liow  l.m«  will  it  lake  lor  the  water  lo  rise  U, , 
n  m  iM  Uin.ahovetheorilke?  llowjnnn  (.)  will  it  take  to  empty  the 
vessel  ?  .»«'.  12  inihes  ;  ta)  67.1*.  sec;  U')  81. 4<.  see.;  1.  (  24-13  "'-i- 

40  In  .1  vortieal  motion  two  circular  lilaments  of  fa<lii  » ,  .  »..  "f  ve- 
locities V,  ,  ;■,  ,  ami  of  c<iual  weiKhl  // '  are  made  to  chani;e  place.    Slw.w 

that  ,1  slahh-  vort.-x  is  pro.h.ced  if  '^'       const.,  and  if  » .      '    ,  show  th,it 

the  surface*  of  e<pial  pressure  ;ire  cones. 

i;o  Sometimes  the  ciest  of  a  >lam  is  r.used  hv 
floatini:  a  stick  /.  into  the  po-ilioii  /., .  where  it  1- 
i«iip|K)rtcd  aniinist  the  verticals.  The  stick  then 
(also!  itself  into  position  /,,  ami  rests  -.n  the 
cresl.      Kxpl.iiii   tlic  reason  of  litis. 

;i.  A  f'-in.  pipe  dischar«es  Hooo  i;,ds,  per  hour 
i.itoii  y-iti.  |>i|H:.    Find  the  loss  of  head  at  the  nine 

ji.  I'lov.ihat  for  a  iiordas  mouthpiece  tunninji  lull  the  coeffuicnl 

I 
ol  tUstharKe  i*    , 


Ft...  7». 


EXAMPLES. 


"5 


5j.    t'iiiii  ihc  (iistharj;e  in  pounds   per    iiimuto    iliii)ii;^h    a    lionla's 
mf)iitliiii<T<.'    I    in.  in  di.iinetfr.  tin-  lip  bcinj;    i:    in.  I)uluw  tlie  waler- 
surface.l.n  when  tlx-  jet  springs  cli'ar  from  iIk-  i'ilj;c,  (/'i  when  the  mouih 
piece  runs  full.  .Im.  (./)  .S1.H45;  <./>)  115.74- 

54.  The  surface  o(  the  water  in  '  tank  is  kept  at  the  same  level  ; 
otitam  tlie  iiiscliarj;e  at  60  in.  below  :he  surface  (di  through  a  circular 
orifice  1  -'|  in.  in  area,  (/'i  through  a  cylinrlrical  ajutatje  of  the  s.iiiie 
si'Clioiial  aiea  litteil  to  the  outside,  (c)  through  the  same  ajutage  fitted 
to  the  insiile,  and  detern.ine  the  iiieclianic.il  elfecl  of  the  elHu.x  in  each 


U 


Alls.  (<f)  4.S5    lbs.  per  sec;  ;:o. 536  ft.-lbs.  per  sec. 

ill)  6.36r> ;  21.404      

(t»  5.4^       ;   13.7-5       1'  runninj,'  full. 

\.(v)       ;   16.63.S      ■■  ■'     "     if  jet  springs  I'lear. 

55.  Water  is  <liscii,iii;ed  under  a  head  of  04  feet  through  a  short  cylin- 
drical mouthpiece  12  in.  in  diameter.  Tiiid  (.M  the  loss  of  head  due 
to  shock,  ^b)  'he  volume  of  discharj^e  in  culm  fiet  per  second,  d  )  the 
eiier>;y   -f  the  issuin};  jet.     [^  =  3:.) 

Ans.  (rr)  20.736  ft.;  (/')  4i"!3  <^^"t'-  ft- ;  '<  !  ^1.04  1 1. P. 
5(1.    If  a  bell-mouth  is  substituted  lor  the  mouthpiere   in   the  prccc-!- 
\n\i  qui  -tioii,  find  the  disi  har^e  and  the  meihailical  effect  of  the  jet. 

,///.»■.  4').2H  I'ub.  ft.  per  sec  .;   ;J4.2  I!.l». 
57.   Compare  tin-  c  tierijies  f)f  a  jet  issuing;  under  an  etfi'ctive  head  of 
100  ft.  throiij:li  (I)  a  I2in.  cvlinilncal  ajuta>;e,  (21a  12-in.  divcrnent  aju- 
tayi'.  (  u  a  12- III.  I  oiivi'ty^ent  ajut.ij;e.  the  an);le  of  1  ohm TLvnce  bein^  i\' 
liraw  the  pl.iin-  of  charge  in  each  case. 

.•/«A.  (I)  3')3.«  n.!'. :  i2)f.72  J.,  n.i'..  ,ji  332.  jS  n.i'. 

5S.  Kind  tne  d  si  harye  tliri)U|,:li  a  rectannular  o|M-niii(i  36  in.  widt* 
and  10  in.  <leep  in  the  virlic.d  face  of  a  dam.  the  upper  cd^e  of  the 
iipeniUi,'  beinv;  10  fl.  below  I  lie  water-surface. 

Alls.  40.2  cub.  ft.  per  see. 

50.  .\  ccntritufal  pump  lias  a  wheel  of  2  ft.  outside  and  I  ft.  inside 
diain..  and  also  a  larye  whirlpool  rliamber.  Draw  to  scale  a  curve  show- 
iiiK  the  pressure  at  different  points  in  the  wheel  and  whirlpool  chamber 
when  the  water  tills  the  pump  but  flows  very  slowlv  tow, irds  the  point  •>( 
discharge.     Take  I  atm.  as  the  pr.  at  the  inlet  suifai  e. 

60.  .\  submerged  sluice  in  the  vertical  f.ice  of  a  reservoir  is  30  (t. 
wide.  The  effective  he.id  over  the  sluice  is  18  inches.  How  lnnil  must 
ilic  sluice  be  raised  to  give  a  delivery  of  45,oihi  e.il   per  minute  .•"    {e  —  .().) 

AHS.  8,  164  MIS. 

61.  The  »ill  of  a  sluice  in  the  vertical  face  of  a  uservoir  is  clear  almvc 
the  lull  race;  the  head  of  water  above  the  sill  is  5  feet.  If  the  sluice  is 
34  ft.  wide,  wh.il  must  lie  the  opening  to  give  i>3,75o  K-''*-  l""''  niin.' 
(f  =  fl  1  .■/«.«.  II.  3  ins 

6;.   A  sluice  in  the  vertical  side  of  a  reservoir  is  partially  submcrgett. 


ITf) 


HX.IMI'UiS. 


the  siirl.K  .•  ui  til,-  lail-i.Kc  walrr  Ik-iiiv;  (,  m,.  ,,l)r.v.-  tiu-  snl.  The  surface 
:.n  thu  iipstKiiM.  SMlr  ,>  .'i  II.  .lix.v.'  Ui<-  mII.  Ii  ihe  sluice  is  iS  l[.  wide, 
wh.it  tnuM  l,c  ihc  t.,ial  ..|.,ninK  .,f  il.c  sluice  •.,,  ^,v,■  i5,63-'  tuns  (uf 
-vx*.  II,>.1  per  h..ur  ^  .,,,,.  ,  ,^,.  f^  _  ^  ,„.,„^   ^ 

63.  Fuui  the  cliM-liari^e  in  cub.  ft.  per  .sec.  llin.ui;li  a  sliarj)  e.lj^e 
..rilice.  6  ins.  sqii.u,-.  in  a  vertical  plate,  llie  centre  of  tile  ..mice  bene  1^ 
ins.  helow  the  water-sui  la<.e.  („)  if  the  velocity  of  appruacii  is  1  ft.Vr 
secon.l  (I  ,f  tlie  channel  of  approach  is  5  ft.  wide  h\  2  ft.  deep. 

,    ,,  A>is.(„)  1.347.S;  (b)  ,.,4. 

04.  .\  r.M-rv..ir  half  an  a(  re  in  area  with  sides  nearly  vertical,  so  tli.ii 
It  may  be  o.n>ideied  prismatic,  receives;,  stream  yielding  .^  cub   ft    pir 
secoiul.  and  .lischarjics  thro„nh  a  sluice  4  f(.  wide,  which  is  raised   2  ft 
Calculate  the  time  lequircd  to  L.wer  llie  suif.ice   ;  ft.,  the  heiul  over 
thcienti.-  of  the  s|.iice  when  <.pened  bcini;  10  ft.  .Ins.  1079  sees. 

05.  Show  that  the  energy  of  a  jet  is>uin-  thruuKh  a  larj;e  rect.uii4iilar 
'.rifice  of  breadth  /I  is  1  ;;/,■,//.,*  -  //}y  //,  ,  //,  i,^.i„^,  the  <Iepths  below 
the  water-suif.ice  ot  the  iipprr  an.l  lower  e.l.nes  of  the  ..rihce.  and  the 
coeiricicnt  <if  di.s.liari;e  beiiin  .f.j;. 

66.  A  reservoir  at  lull  wat.T  has  a  .ieptli  ,.1  40  ft.  ..v.r  the  centre  of 
the  dischai.i;in.--sluice.  which  is  rc.taiinular  and  24  in.  wide  by  iS  in. 
deep.  I-ind  the  dischar-e  in  cubic  feet  per  second  at  thit  dei'.th,  ami 
also  when  tli.'  w.itcr  li.is  fallen  to  30,  M.  and  10  ft.,  respecfvely  ;  lind 
the  mechanical  ellect  of  the  elHiix  in  each  case,  c  beinj,'  .6i;. 

.///«.  94.S   cu.    ft.;  Si.  I    cu.   ft.;    67   cu.  ft       47.4   ,u      ft      j;i    > 

H.l»..  2,SoH.I'.:  ij-.sM.P.  ;  33.9511.1'.  '  ■ 

67.  Hcpiirelhehead  mcessary  to  >;ive  7.8  cu.  ft.  per  sec. nd  llirouRh 
■  in  oritire  30  s(|.  in.  m  sectional  area.  ,   bein).j  .0.'5.  .  l,„,  3.S.,,  f, 

6.S.  The  upp.r  and  lower  edj-cs  of  a  vertical  rectaiiiju' ir  orifice  are 
'.  and  10  ft.  below  the  surface  of  the  water  in  a  cistern,  respectively 
Ihe  width  ..t  the  ..nli.e  is  i  ft.     Find  the  .lisch.irRe  through  it. 

•''"•  5<'-4-  <"u.  ft.  [)er  .sec. 
'»>     I  he  two  slui.es  ca.h  4  fi.  wi.lc  by  2  ft.  deep  in  a  lo<  k-^r.ite  are 
s.ibmerne.l  one  half  their  depth.     The  ...nstant  liea.l  of  water  ab.)ve  the 
axis  of  th.'  .sluue   is   12  ft.      Fin.!  the  discharge  ihiou^-h  the  sluice,  the 
velocity  of  appriMch  beinn  4  ft.  per  secon.l,  ,   beiuf;  .625. 

■-''"•  i6.6-''.2  cu.  ft.  per  nun. 

~j.  find  the  (low  through  a  s.piaie  openinj;,  one  diaj;onal  beinn  vei- 
ti.ai  and  12  .11.  m  leiiyth.  the  upper  cxtr.-mily  of  the  .lianonal  he,„y  m 
Ihe  .Hiirfa.e  of  the  walei.  and  *■  hpiny  .62;.        .///,.   1  7:4  ,„.  f,    ,„.,  ^,.^. 

71.  To  liiKl  the  f4iianiitv  of  water  conveyed  away  by  a'canal   ^  ft 
wide,  a  board  with  an  01  nice  2  ft.  wi.le  and  1    ft.   deep  is  placed  a.  1. -h 
the  canal  and  dams  it  back  until  it  attains  a  heiuhl  of  :{  ft.  ab.ive   ih. 
bottom  and    lift,  above   the  lower  ed^e  ol  the   orifice.     Find  the  .1  ^ 
charge  in  cubic  feet  {kt  second,  <  beinK  .'m,,. 

.;«'    i7  5<;<>r2o2i  if  orifice  IS  drowned. 


i  : 

"i 


I 


1 


EXAMPLES.  '  '  7 

73.  Six  lliousand  };iillon<  of  w.ilei  per  minute  arc  foicc<l  thruu^li  a 
line  of  pipuij;  ./A'C'aiid  arc  disi  li.ir),'iH]  into  the  atinusi>liere  at  C.  wliicli 
is  6  ft.  veriK'.illy  aliovc  .1.  The  pipe  .1/!  is  6  in.  in  iliaim-tcr  .md  1:  ft. 
ill  leiimli;  llie  pipe  A'c' is  12  in.  in  ili.inutcr  ainl  iJ  It.  in  h'ligth.  D\— 
reoardiiii;  friction,  linil  tlie  "Uiss  m  >h<ii  k  .iml  draw  tlie  plane  I'l 
cliaiiie.  .h.'s.  l.iis.s  I  if  held  ill  shiiik  —  58.3  ft. 

7-;.  Wh.it  (pi.iiititv  <if  water  tliws  thri>ui;h  tlie  vertiial  aperture  o(  a 
ilam.  Its  widlii  i.iiti);  },(>  in.  and  its  ilrptli  lu  in.  ;  the  nppi  r  c  dye  nf  tin.' 
ii|K'rlure  is  i<>  ft.  beli'W  liie  sulfate.  .h/s.  50.(15.  i  11.  ft.  per  s<(:. 

74.  264  1  u.  It.  of  \vai<'r  are  disciiarj{ed  tiirouj^h  an  urihei;  of  5  si) 
ins.  in  j  niin.  10  sec.      Fin<l  tlie  meat'  «,  l,„  ,m   ..f  .fllnx. 

,  his.  fi4  ft.  per.  sei'. 

75.  One  of  tlie  locks  on  the  Lariiuir  (  .111.11  li.is  a  superticial  .iie-i  of 
ahout  IJ.150  SI),  ft.,  and  tlie  dilfeience  of  level  between  the  surf.n  es  of 
the  water  in  llu'  lock  aiKl  in  the  upper  reach  is  9  It.  Kach  leaf  of  the 
j^.ites  is  supplied  With  one  sIuk  e.  and  the  water  is  levelled  up  in  2  min. 
4,Ssecs.  Determine  the  proper  aiea  of  the  sluice  opening.  (Centre  of 
sluice  JO  ft.  below  Miif.ice  of  ujiikt  re.ich  and  c  —  .025.) 

.  //;>.  Area  of  one  sluice  =  43.39  srp  ft. 

-ii.  The  liori/ontal  ~eitioii  i.f  a  lock-rli.imbrr  may  he  .issumed  h 
re(  taiiv.i<-.  the  lenj^th  l.ein;..;  3' o  II.  When  the  <  hamber  is  full,  tli'-  sur- 
faci'  widtli  between  tin  side  w.dN,  which  have  e,;ch  a  batter  of  1  in  12, 
i^  45  ft.  Ilow  loni;  wiU  i!  take  to  empty  the  lock  through  two  sluices  in 
the  nates,  cieh  8  ft.  by  2  ft.,  the  lieinlit  of  the  water  uIhjv  the  centre  of 
thesluiii-s  bein^;  13  It.  in  lhe!'"k  and  4  ft.  in  the  canal  on  the  down- 
stream Side.  •'"'•  f'OO.rj  -e(   ,  .    beillK  .(>2S. 

77.  Water  approaches  a  le.  t.iiij^ui.ir  openni;  1  It.  widewillia  Velocity 
of  4  ft.  per  second.  At  the  opmiiin  the  head  ol  water  o\er  the  lower 
i-di-e  =  13  fi..and  over  tlie  suif.ice  of  Ihr  t.iil-rac  e  .  1  2  It.  ;  the  ilischarjje 
through  the  opening;  is  70  cii.  ft.  per  second.  Tind  the  lHi).;lit  of  the 
i.penini.'.  c  lieini:  .(iJ).  •'■"'•  '•'■'•-  fl- 

-iS.  The  wate:  in  a  le^^ulalinn-ch.imber  is  8  ft.  below  the  level  of  the 
water  in  the  can.il  and  8  ft.  above  the  centre  of  the  <li.si-har';in^:-sliiice. 
iJeteimine  the  rise  in  the  <an.i!  which  will  iiu  rease  the  dischar^'e  by  10 
lier  cent.  •■''"•  "•'  "'  ft- 

The  hori/.oiiial  sectional  ana  of  the  chamber  is  constant  ami  eijiial  Uv 
...lo  s(|.  ft.  ■  in  what  time  will  the  water  in  the  chamber  rise  to  the  level 
of  lh.it  in  ill'- canal,  if  the  disch.irKinn-sluice  is  closed;  the  sluice  be- 
tween the  canal  and  chamber  lM;in!L>  3  sq.  ft.  in  area?    .his.  150.S3  si-c. 

7iy.  .\  lock  on  the  Lachine  Canal  is  270  ft.  lonj;  by  45  ft.  wide  and  has 
a  lift  ol  8|!  ft.  ;  ', here  aie  iwosluucs  in  each  leaf,  each  8J  ft.  wide  bv 
?i  ft.  deep,  tlie  head  over  the  liop/onlal  c:cntre  lim.  of  the  sluices  i.s. 
K)  ft.     Find  the  lime  n.pnreii  to  hll  tlie  loc  k.  ./«v.  163.5  """<  • 

80.  The  locks  on  tlie  Mont>;onRiv-liiic  Canal  are  81  ft  lonj;  and  7J 
'•    wide,  at  one  of  the  locks  the  liH  •  i4'ii.  pijie  leads  the  water 


iiS 


fX.IMPl.HS. 


from  tl\L-  iii'iHT  li'vel  ;.iul  .lisrli.iri,>cs  below  tlip  -iiirf.  ce  oi  the  lower  level 
into  tile  l(i(  k-(liainl.ei  ;  tliemoulli  of  the  pipe  is  si|u.ii<-,  :  ft.  in  the  side, 
and  i;r,iMii;iliV  i:i..in;es  into  a  (  inular  pipe  2  it.  in  diaiiietei.  I'lnd  tiino 
of  linin(,'lhe'l...k.      ..    -^  :.»  W //v.  1.^2.11  see. 

Ki.  A  i.iiial  lo(  k  i>  115.1  ft.  Ions;  an<l  30.44  ft-  wide;  li:e  veriie.il 
deptli  from  rentre  of  s'.uiee  to  lower  reach  is  1. 0763  ft.,  ''e  cliai!;.'  being 
'•.3i>45  It.;  tile  area  of  the  two  sliiiees  is  2  x  6.766  S(|.  ft.  Find  the  tune 
of  miiiif;  lip  to  I  eiilK-  of  sluices,  u'  =  .625  lor  the  sluice.  l)ut  is  reduced 
l.»  .54.S  when  both  are  opened.'     AUo.  tind  time  of  lillim^  up  lo  level  of 


upper  reach  from  centre  of  slnice-iioor^. 


.hi.\.  .';  sec.  ;  39s  sec 


■S2.  lion  many  i;^dl  Ills  o(  water  will  flow  thron^li  a  >/)  notch  in  24 
liours  if  ihe  depth  of  the  water  is  27  ins.  for  the  lir-,t  S  hours,  12  ins.  for 
the  second  S  hours.  .11  il  ',  ms.  lor  the  thud  S  hoiir>,  .   beinp;  .(<? 

,l>n.  3.'i74.40"- 

Sv  Show  tlia!  Ill  ,1  I  li.iiiticl  of  V  sedion  an  inclement  of  10  pel  .  1  nt 
in  the  <ieptii  will  pn-dniea  coi  rer.pondins4  increment  of  5  pet  <  cut  in  tlic 
veloiiiy  of  Mow  aiul  ..|  25  per  cent  in  ih.'  di-cli  iii;i'. 

N4.  The  .inj-lc  of  a  triangular  nolcli  i-^  •h.'°-  How  '"'k''  must  the 
water  rise  in  the  ii-itdi  so  that  tiie  di>cliai!;e  may  be  ickki  -allons  per 
„,j„,,,,.  5  ./;,■>.   12  ins   very  ne.irlv. 

S5.   .\  reservoir.  rect.ini;iilar  in  plan  .ind    io.ikxj  sip  ft.  in  aiea,  h.s  jn 
ont    side  a  oo"  irJaiiRiilar   iu>lcli    2   (I.  deep.      It  tiie  reservoir  is: 
what  time  will  the  'cvel  sink  6  ins.?  .his.  4i/>.^7  >' 

.S6.  How  Inn-  will  It  l:'ke  to  l.iwrr  bv  3  ft.  the  surface  of  a  reservoir 
of  (<4o.ooo  s<i.  ft.  are.i  tliiou-h  a  '/>"  V  notch  \  It.  dcep^ 

.Ins.  40.  ;o  hr-..  r  beini;  .'•. 

.S7.  iMiid  the  di-charn'-  in  >  nine  feet  pel  -.eii.nd  lhroui;h  i  '>o'  notch 
wh<-n  the  ilepm  •''  'v.i.-.   i^    •-'<',■  i.mI,  ii  i-  4  ft..  .   bciiii;  .r.17. 

.///>.  >'4.24. 

SS.  .\  ponit  Who-.-  :ii<M  1-  1  -..i.,!  Mj.  i;.  n.i-^  ..ii  ovei  fail  outlet  36  in. 
wide.  wlii>  h  at  the  ommeii.  i meiil  i.f  the  dischaiKL-  I'.a-  a  head  of  2.8  ft. 
l"ini|  ilie  tune  recpiiretl  to  lower  the  surf.ice  12  in.  .hn.  3}4-3'^  "ec. 

S.,.  II. w  miidi  water  will  tl.iw  lhroui;h  a  ifctanRiil.ir  notch  24  in. 
wide,  the  surface  of  still  water  beini-  8  in.  above  the  crest  of  lite  notch  ? 
(Take  into  account  side  contraction.!  ./'/>.  3  38?   cii.  ft.  per  sec. 

^to.  A  wen  pa^ses  6  cubic  feet  per  second, and  the  head  over  thc<  h  -t 
).i  .S  inclu-..      rind  the  leilKth  .)f  the  weir,  .   bi-ini,'  .62;. 

-/«.«.  3.31^)8  II. 

<)l.  A  wi  ir  40.)  ft.  lon«.  with  a  v-i"-  depth  of  w.iler  on  it,  discharges 
thi.'iudi  .1  louii  weir  ;oo  ft.  loiiij.    l-'iiid  the  depth  of  water  on  the  latter. 

,///*.  .6457  ft. 

•  (_'.  .\  Hill  1-  it.>  I  I'lii^.  how  liiKli  will  the  u.ii.M  rise  over  it  when 
it  rises  .6S  fi.  upon  an  iijip-r  weir  750  ft.  1  nm  -  //«(.  .8413  ft. 

03.  What  sliiiuld  oc  the  lieiKht  of  a  iliowne.l  w.n  400  ft.  loni;,  t.. 
deepen  the  water  on  the  up-slreain  side  |)y  50  perienl.  the  section  ..f 


r.x.i.uri.FS. 


1  10 


X\m  stn-iitu  beinu  40.0  ft.  x  ,s  fl..  anci  uio  vclucity  of  api.t-a.  1.  ;,  It.  prr 

.  5  '  .Ins.  7.ob4  It. 

94    The  (Icptli  '>f  w-iler  on  tlir  <  p.-i  uI  a  rectangular  notch   5  It.  lon^; 

is  ::  feet,      rind  tin-  discliar-c  wlu'ii  Hie-  nutcli  lias  ^.n  tw.>  end  contrai  - 

li.>ns,(/.)..nee..dc..ntracti..n.(rM....-nd  contraction.,  in  cell  case  lHMn^' ^ 

.///.«.   ,M    J.V'/xj    <"•    ll.    per    sec:   lAi    4J--54    '  n-   "•   P^-T    ^'■'•- 

(,)47.i4'i'.  fl.  persec.  ,       ,     ,  „      ■ 

y:  Sliowtliat  upona  weir  10  It.  i.ni.uwiih  12  ms.  dontli  ot  watei  (lovvnii{ 

over,  an  error  ..f  ,,,'„„  of  a  f"<>'  '"  "'c-aMuini:  liie  head  will  cause  an  error 

of  3'cu.  fl.  per  Miinule  in  the  <li>.  liar,,'e.  and  a'l   en..,    ,,f  ,,>,„  of  .1  f....l  ni 

ni-.isnrin«  the  lenulU  of  ti;e  «i  11    wili  cause  an  en..r  ..I   .:  cti.  ft.  in  the 

discn.irye.  •     ,  •   ,  , 

„(..  In  the  w.-M  at  Kiiraloe  tl.e  ftal  length  is  lioott.,,>l  which  77911. 
fr,ni  tlie  east  ahutnieiil  is  le\el.  while  the  remainder  slojies  1  in  214. 
.4,\iii-  aK.lal  rise  at  tiie  west  ahntmenl  <.l  1.5  It.  C'alcnlate  the  total 
d  IS.  liar-cover  the  weir  when  the  depth  of  ^\.iler  (ni  the  level  [.art  is 
l.S  ft..'wllich  uivfs  .3  ft.  on  hii:hest  part  of  weir.  lUivide  slope  int.. 
8  lengths  .if  40  It.  each,  and  assume  them  severally  U'vel.  with  a  hcati 
e.niaU..  the  arithmetic  mean  ..I  llie  head  at  the  heiiinnin-  and  en.l  of 
each  length.)  . /./.v.  7+-/"  «'■  ft-  l»>^f'^^- 

97.  .\  watercourse  is  t-.  he  augmented  hv  the  ^treams  and  sprm^js 
.il),.ve  its  level.  The  latter  ai.-  sewrally  d.iinin'd  up  at  snital.lc  places 
.ind  a  narrow  h-.aid  is  provid..l  in  whu  11  an  ..p.nini;  \:  in.  lon^;  by  <i  in. 
.leep  is  <ut  for  an  ..verfall;  it  w.t-  s.irmi>ed  that  this  would  he  sutlicient 
|..r  the  lurnest  streams;  an..th<-r  piece  attached  t.>  the  former  would 
redu.--tlie  h-ni.;tli  to  '.  m.  for  smaller  streams.  Call  iil.ite  the  delivery  hv 
■ilie  following  sticams  : 

in  No.  I  stream  with  the  i:-in.  notch,  depth  over  crest 
"   No.  2  "        "      '•  "'• 

•■   No.  3  '■     '-'-"'•       ■' 

••  No.  4       ('-'"• 

(Take  into  accunt  the  sid.-  c.mli.K  ti.ins.) 

.Ins,  No.  I.  .6<;6cu.  ft.;  No.  2.  .3(15.^  cii.  ft..   N.).  3. 
N.).  4.  .1275  cil.  ft. 
<„S,  .\  rectangular  notch  h.is  tw.>  complete  end  contractions  and  the 
lenntli  ..f  the  crest  is  //inr  limes  the  depth  of  the.  water  on  the  crest, 
Wh.il  must  he  llie  leiii-lli  of  the  crest  to  jjivc  a  minimum  disch.irf.'e  <jf 
1S.750  (;als,  per  minute,  <   lieiii',;  ;  ?  -'«'•  S*/  ft, 

9.).  .\  stream  30  ft.  wide,  3  ft.  ile.-p.  <lischarj,'es  ;to  cu.  ft.  per 
siiond;  a  weir  2  It.  deep  is  hiiilt  ;icross  the  stre.itn.  F.nd  increased 
depth  of  latter,  un  nei;leclini;  velocity  of  approach,  (.')  t.ikini;  vcl..city  of 
approiich  into  account.  .his.ia)  1.26  ft,  to  i.-'d;  ft.;  ('•)  1.19  ft. 

10  >.  In  a  stream  50  ft.  wide  and  4  ft,  deep  water  flows  at  the  rate  of 
100  fl.  perminute;  lindthe  heisjlit  of  a  weir  which  will  increase  the  depth 


■  .37  ft. 
-.41ft. 
=  .29  fl. 

-  .10  ft. 


.49-- 


'u   ft.; 


I  20 


ix.-iMri.iis. 


Ill  ()  ft..  (I )  II cj» lectin^'  velocity  of  iipproacli,  12)  takiniL;  vclocityof  approacl. 
inli)  account.  ./'■>■.(  I  1  4.41  26  ft.;  (2)4.4305(1. 

101.  A  sireain  50  ft.  vvi(  e  and  4  ft.  ilci-p  has  a  vtlo(  ity  of  5  ft.  p'.r 
^1  coiid  ;  liiid  the  lici!;lu  of  the  wen  wliich  will  double  (lie  depth,  ilt 
nejjiectiiij;  velocity  of  approacli,  1  Jl  l.ikin^  velocity  of  appnjach  into  a<  - 
count.  .''/>.  Ill  5.651  ft.;  (2)  5.6.S62  ft. 

102.  A  ^tle.lIn  So  fl.  vviiic  iiv  4  ft.  dei-p  <ii-;chart;es  acro.-^s  a  vertii.il 
section  at  the  r.ite  (jf  040  1  u.  ft.  ptr  seconii  ;  a  wen  is  l)uilt  in  the  stream, 
incieasMij;  its  deiitli  to  6  ft.      Kind  the  heii;ht  of  the  wen. 

A  IIS.  4.233  fl. 

103.  Salmi  ni-yaps  are  constiucted  ni  .1  weii  ;  tln'y  are  eacii  10  ft.  wide 
and  their  crests  are  iX  in.  below  the  weir  (  rest.  Calcnl.ite  tile  discii.ir;;e 
down  three  of  these  yaps,  the  water  on  the  level  part  il  tne  weir  l«  in.; 
S  III.  deep.  .hii.  23S.  I  3  (  u,  ft.  per  sec. 

104.  A  channel  of  ie«  t.injjniar  se(  tioii  and  20  ft.  wide  convey.-. 
i.600,000  i^ail'iis  per  hour,  the  deplii  of  the  stieani  heini;  S  ft.  A  dam 
2  ft.  Iiii;h  is  iiiiilt  across  the  (  liannel.  Kind  the  •  lieit^lit  of  swell  "  (1.')  dis- 
ren.irdnii;  the  velcjcily  of  approach,  ((*)  taking  the  Velocity  of  approai  h 
into  accouni.  .-Ins.  (.«)  .07  ft.;  {6)  .0545  ft. 

105.  The  water  in  a  flume  S  ft.  widir  is  3  ft.  dee[)  and  is  supplied  from 
a  sluice  6  ft.  wide  at  the  i.ite  of  27.000  i;a!s.  per  iiiinute.  If  tile  coi  lii- 
cient  of  conlracti(Mi  is  unity  .iiii'  il  lo  percent  is  allowed  for  fraction. i 
loss,  find  the  (inference  of  level  between  the  water-surf.ices  above  tin 
sluice  and  in  the  Ihiiiie  when  the  -hiice  openiii};  is  o/)  I   ft..  1/')  2  ft. 

.his.  (.1)  2.32  It.;  [i)  .31  ft. 

106.  A  stream  of  icctangiil.ir  section  24  ft.  wide  delivers  145  cu.  ft. 
per  second.  The  erifje  of  a  drowned  weir  is  15  ins.  below  the  surface  of 
the  water  on  the  down-stream  side.  iJeteiniiiie  the  ditlerence  of  levci 
between  the  siuf.ices  <;(  the  water  mi  the  up-  .nu!  down-stie.im  sides,  the 
velocity  of  approach  bciny  ;  fl.  per  second. 

Ans.  ,  .9  ins. 


(  iiAn'i-K    11. 

I'LriD    IKiiTloN    AMI    IMl'l.    ILi  i\V. 

I.  Fluid  Friction. —  Tin-  tirni  lliiid  friction  is  apiilii-il  to 
tin-  II -istjuuc  ti)  inruion  wliicli  i--  (l.vclopi-d  wliui  a  tlufd  flows 
ovir  a  solid  surface,  and  is  dm'  to  tlir  viscosit>-  of  tile  fluid. 
This  resistance  is  uecessaril}'  acconi]).inied  1)\-  a  loss  of  eiier;4\- 
caused  1)\-  the  i>ro(hictioii  ol  eddies  aloni;  the  surface,  .ind 
similar  to  the  Ins-  which  occur-  .U  an  ,d)riii)t  chan-e  of  section, 
or  at  .in  anL,de  in  a  pijie  or  chauiul. 

j-'roude's  exiKriments  on  the  resistance  to  the  <(ii;euise 
motion  of  planks  in  ,i  lluid  mass,  tiie  i)lanks  heiuL;  ,■",,  in.  thick, 
lo  in.  ilee]).  and  I  to  30  ft.  loni,'.  each  plank  ha\in;j;  .1  fme 
cutuater  and  run,  are  sinnmarizod  in  the  follow  in;,;  t,d)le: 


I.tiiK'li  "f  Siirt.Ki-  in  l-'i 


N.inirr  ()(  Siirfate 

C'n-ftini,' 


J  Kci-t. 


H  :  I 


II 


II 


V.iriiish J.o<>i  .4i!.3<P''''5  -325  ■-(>*  ''Sf  .87^ 

l'..i,ifliiic I  .3S  .370i.<)4    SU  .I*'"!.')'  -ST' 

Tinfoil 2.U}  .30  .2i>5  l.c)<)  .2TS  .-'fi?  l.qo:.a()2 

Cahc., 'l.r,\  .87  .73?  t.-)-.'  .;>2i  .S'M  t  ..'•(i  .  JTI 

I'liir  satiil    ..J.""  .81  .(hji)2.<>.  .e."?  ^'"S.""    ^i'O 

M<Mllum  sand. .    ..     .•.i)»  ,cjo  .710  3..«>  .dJf  .4^8  2.1K    .^34 

Coarse  sand -•.<io  I .  lo'.."-  "'    -   ' 


H       c: 


24"  1.63  .2511 


244  1.!^?  .24'' 

.2ia 

4»7  l.fcT  .47J 

.4-3 

3S4  l.iV'  .411: 

•337 

4(it  J.iiit  .4!>» 

.450 

.)  :n 

Columns  A    ;^i\e   the    jKiWer  o|   the   speed  1. 
resistance  is  approximatel)-  proportional. 

('ohimns  U  i^i\e  the  mean  rcsistanci-.  in  pouiu 


t"  w  liK  h  the 

is  ]ier  square 
121 


122 


iLum  ii:icrio\: 


foot,  of  tlic  uliolc  surt.icc  of  a  hoard    of  ilu-   IiMit^tlis  stated    in 
tlic  tahlc. 

C'oliinuis  C  ;4ivc  till'  resistance,  in  pounds,  of  .i  square  foot 
of  surface  at  the  distance  sternward  '"'dm  tlie  cutwater  stated 
in  the  headin:^,  each  iilank  ha\in:4  a  standard  speed  of  lo  tt. 
per  second.  The  resi:-tance  at  other  speeds  can  he  easily  cai; 
culated. 

An  eNannnation  of  tlie  tahle  show  s  tliat  tii<-  mean  resistance 
])er  scjuare  foot  diniinislies  as  the  ieni;tli  of  the  pi. ink  increases. 
Ihis  niav  \)c  eNi)l, lined  by  the  supposition  tliat  the  friction  in 
the  forward  portion  of  the  pla.'^  levelops  ;i  force  wliich  dr;i!:;s 
the  u.iter  .iloni,'  witli  the  surface,  -  >  th  it  the  rel.ilivi-  velocit>- 
of  tlow  oxer  the  rear  portion  is  diminished.  A^ain,  the 
ilecre.ise  of  tlie  mean  resistance  per  s(|uare  foot  is  1  '^J  lb.  when 
the  leni^th  of  ,i  varnished  pi. ink  is  increased  troin  j  to  20  tt., 
whih'  it  is  i.nlv  .02S  lb.  when  the  leiv^th  increases  tidiit  Jo  to 
50  It.  liince  for  -re.iter  lengths  th.m  50  tt.  the  decrease  of 
resistance  m.i\    be  I'isreL^arded  without    much,  if  .in>-.  [iractical 

effect . 

'Ihus.  .4rner,ili\-  s])eakini;,  tliese  e\periments  indicate  that 
the  me.m  resist, mce  is  proi)ortion.ii  to  the  itth  i)ovvcr  of  tlie 
lel.itive  velocity.  //  varyin!.^  from  I. .S3  to  J.K),  and  its  ;iverat;e 
v.iliie  beini;  \  er>   ne.irix   2. 

Colonel  In Miifoe,  as  a  result  of  e.\])eriincnts  at  Deptford, 
also  assumed  tli  mean  resistance  to  be  i)n>pi)rtional  to  the 
•vth  power  ol  the  lel.itue  \elocit>-,  the  value  of  //  in  three  series 
of  observations  beini;  i.'i'>.    1.71,  and   1.9. 

rile  Iriction.d  ri'sistancc  is  evident!)  jiroportional  to  some 
timclion  of  tlie  velocity,  /•'(.■•i,  which  should  \  anish  when  ,•  is 
nil.  as  when  the  surface  is  level,  and  should  increase  uitli  ,■ 

Coulomb  ,1-sunied  the  function  />  1  t^  be  of  the  form 
„-.  .1_ />;■-',  ,;  and/'  bein^' coefllcietits  to  be  deterniiiiel  b\  ex- 
periment. I-'.\perinient  siiows  tliat  when  /'  iloe-  not  exceed 
;  ft.  |)er  minute  the  resistance  is  directiv  proportion.il  to  the 
velocitv .  but   th.it   ii  is  more   nearlv  propurtion.il  to  the  scju.irc 


i 

I 


III  in  I Kicriay.  i-.) 

of  the  velocity  whni   th.    n  .loiiiy  .n.loK    ;,o   ft.  \>cr   minute; 
•ir. 

/•u  I        -/.-  when  .-  ;  tt.  I'll-  niimitr. 


/•I  .-I  =^ /;  ■  uluii  ,•        .V'tt.  ]nv  nii!\utf. 

.\u,Min.  >ihs(.rv;ition-<  en  Uk-  tlow  <i|"  \vat>  r  in  t'>\vii  mains 
niilicati-  that  n..  .liltcnni  i  ..t  rt-i-laiK.'  i-  .l.\cl..ii<-.l  luulcr 
.vidfly  varyini;  ])rcssurc^.  antl  this  iiuki)LiiLkni.i.'  ■  t  jin-isurc  is 
..N  .  \.rifiO(l  In-  (niihimh-  i-xiKiinuMit  -howin-  tliat,  it  a  tusc 
;  ..-.ullatcii  in  uatii.  tlu;n-  i-  n"  ai)i)aiinl  cliaii-c  in  tlic  rate 
.r  .kcreaSL-  ..(  tlic  osi  illation-,  wlutlur  the  water  w  umler 
tmo~i)herie  pres-ure  or  not. 

Irom  thi    preifdin;.;  an.l  otiur   -imilar  expennuiils  the  lol- 
.winLi  -eneral  law--  of  iluiil  Iriition  liave  heeii  toinuilated : 

I     '!  he  iVivtional    .e^^istaiue  i-  independent   of  the  pressure 
rtween  the  tluid  and  the  -urfaie  o\  er  whieh  it  flows. 

; ■,    '\'\'[r  f,-i,  t'"".!    vi-i-t.mi.     i-~  proportional  to  the   area  oi 

the  surface. 

?     The   tVietiona!  re-i-tanee  is   proj)ortioii,d  to   some  fune- 
•011.  u-nall\-  the  Mjuare,  of  the  veloeity. 

I.,  these  tliree  laws  ma\-  Ik-  added  a  fourtii,  \i/..  : 
4     The  frictiona!  resi-tan^e  i-   proportional  t,.   tlu   den>it\- 
iid.  vi'-cosit_\-  of  the  t1ni.! 

.\  nt'th  lau.  \i/...  that  --the  trutionai  i<-i-laiui  i-  inde- 
jK  ndeni  of  the  nature  of  the  -urf.u  e  a-ain-t  whiih  the  nui<l 
•lows,"  has  been  scmutinu-  enunciated,  .md  at  ver>-  low 
elocitie-  til.  lau  i-  ai)i)ro\iniatt  i\  true.  ,\t  liiL;h  velocities, 
,. owever.  such  as  are  common  in  enL;ineerin|,^  practice,  the 
I.  sjstaiice   has   been    sl,,)\\.i    l>\   es]ieriment,  and   especiallv  hy 


Ml 


e    e.xperiinetits    i  .irried    out    li.\-    Darcv,    to    he    \  er\-    lart^ely 


influenced  hv  th<-  nature  of  the  surface. 


i^  >  /•/.;  in  I  Kiciiox. 

lA-t  />  he  till-  fiictimi.il  legist. nui-  in  pnuiiil-^  \>v\-  -.iinarc  foct 
ol  surliicc  at  a  xclocity  ><\  i   It.  ])(r  scciiid. 

Let  ./  he  the  area  nf  the  surfaec  in  '«|uare  feet. 

I.et  :■  he  the  relatixc  selocity  of  the  -surface  aii'l  tlie  water 
in  whii  li  it  is  inunersecL 

I.et  A'  })e  the  total  frietional  resi'itani.  e. 

'liien  frtiin  the  laws  of  (luiil  friction 


■Jake  /  =    ■■■"■  />, 
Then 


l)(.iML;    the   speeifie  weis^ht   "I    the    ihiid. 

A'       -    /-TcV/--. 


I  he  coefficient  /  is  ajjpro.xiniatelx' constant  for  any  Ljiven 
surface,  and  is  termed  the  toefficieiit  of  tinid  triction.  The 
])o\ver  ahsorhed  !)y  the  fr/    ional  resistance 


TAIM.K  t.IVINC.  lUK  .WKR.Ai.i:  V.\  l.ri'.S  OV  /  IN"  IIH".  CASE  ol- 
1..-\Ki;K  SfKlACKS  .MOVINi.  IN  \N  I  N  1  )i;i  I  N  11  1.1.  V  l..\K(,K 
M.XSS    OK    WAIEK. 

Surf. ICC.  CueHicietu  nf  Frictiun  (  /  ). 

New  \\ell-])ainti'<l  iron  pl.-ite 00489 

Painted  and  ])laned  i)lank 0035 

Surface  of  iron  ships 00  V'3 

Varnished  surface.         OU25.S 

l-'ine  sand  surf.ice  004  1  .S 

Coarse  sand  surf.ice "O503 

|-'.X.  The  wetted  surfiiee  of  a  vessel  rnovine;  at  S  knots  per  hour  is 
7500  s(i.  ft..  ,-111(1  the  ••esislance  is  .4  lbs.  per  sq.  ft.  at  a  spefd  of  10  ft.  per 
seeoiui.  F-"i!iil  the  surface-resistance  and  llio  horse-power  retiuired  U^ 
|)ro|icl  the  vesicl. 

The  rtshtaiii  e  111  lbs.  per  S(i.  tt.  at  I  ft.  I'Cr.  see.  —    '•  „  =  . 


t-l.UlD  IKI'.TIOS.  125 

Tht-rcfore  the  l(>l<i/  ,itn-ri\ui/,iiii  ,• 

-=      ^     .7500    ^     ---^     =  i^^7\  ll's. 
1000      ■^      \  to   X  60/ 

,.,       ,  „  «   '^  ''««''  ' 

1  lio /;i»r>e'-/.ijiv>-  —  5487.3   y  -       -    •         -         1  U- (J. 

2.  Surface  Friction  of  Pipes.  A-Mimiii'4  tli.it  tli'- l.iws  of 
fluid  friction  alrcaih'  cmnKiatcd  hold  t^ood  ulun  water  flows 
throuLiii  a  ])iiK-,  it  lia^  boon  -.liown  h\-  mimoroo  ox|)orimo!its 
til, it  the  coofficiont  of  friction  /' lio>  hotwcon  tlio  liiin'ts  .005  and 
.01.  it>  avcra;4o  \aluo  iindir  oidinar)' condition^  boinj;  aliout 
.0075.  No  sini^lo  \aluo  of  /  is  applicahlo  to  \or>-  difforont 
case-;.  Indood,  / doponds  not  onl)-  u])on  the  condition  of  tlio 
surfaco,  but  aNo  uixin  the  tlianieter  of  tho  pi|)o  and  tho  \elocit\ 
of  the  wator.  Some  authorities  ha\'e  cxjjre^sotl  its  \.duo  by  a 
relation  of  the  torm 

/■  .     ^' 

It  and  /'  boinL;  constant--  wIm^c   \.ilue>  arc-  to  be  determined  b\- 
exporinicnt. 

Tho    follow iiiL;    talile    '^ivr^    >oiuo    of   the    host    nuniorical 
results  obtained  fe.r  </  and  /' : 

Aiulioritv.                                   ''  '' 

I'rony ooo_'iJ30  .00003466 

!•  Aubuiss.jii             00020i>0  .000037608 

i'Atolwein. OOOi-Osi)  .OOOO4441 

In  pipes  of  sin, ill  di.uncter  in  uhieh  the  \olocit\-  of  flow  is 
le-is  than  4  iiis.  per  >eeond  the  term  ./  m.iy  be  disrcejardctl  so 
that 

/  ^^' 

J,        J.  • 

In    ordinary  piaetiei-    .Uid  \\iieh  tiii    pipes  h,i\e    beiii  in  use 


1  "Ct 


si'Rt.-ich:  f-RfcTiox  1)1-  rtrF.s. 


fur  "-ijiiH-    tiiiK-   tlic  \clocit_v  u-u, illy  lxciu-iU   4  in-;    per  -^ecotiff, 

/> 
iiiul  tin;  iciin        iii;iy  tlini  he  iiisrc;4ar(l>'il,  -u  tli.it 

/ 


1  )ari  y  s  atul  ntlirr  more  rtci  lit  r\-i)i  riiiu'iu-;  show  that  a 
.mil  /'  arc  not  mii^taiit.  hut  an-  iiniri'  lorrrctly  expressed  a-> 
fmutioi'-  of  thr  (liaiiutii  In  1  )arc\  -  experiments  tlie  pi|x--. 
were  laid  ver\-  nearly  hori.yonlal  ami  the  j-.^ad  eould  Ix  varied 
at  will  hy  the  opeiiiii;^  or  liosiiiL;  of  \ahe->. 


^^ 


/'/IH. 


JL 


—  lOt 


i-- 


-1«-- 


19.1 


Fi.;.  7.,. 

I'irznmetcrs  v.cre  inserted  .it  int<r\.ils  of  iri4  ft.  50  m  ), 
commencing,'  at  15.4  It.  14.7  in,  from  the  inlet,  i.e..  at  a 
point  wliere  the  pipe  was  running,;  full  and  the  tlow  was  steady, 
rill-  iii)per  eiiiis  of  the  pie/ometers  terminated  on  a  vertie.il 
pl.mk  sf>  placed  .is  to  allow  tlie  water-levels  in  them  to  he 
«il)ser\ed  and  compared.  In  any  two  conscc.itive  pic/.ometirs 
tile  difference  of  level,  which  is  of  course  constant,  represents 
th(  friction.nl  loss  of  head  in  .1  ir.4.ft.  Ien;,'tli  of  pipe.  I-Vom 
the  results  of  these  ex|KTiinents  Darcy  m.ide  the  following' 
«leductions : 

[(I)  I  hf  ln>iniun  iiii.\/,iiii,  ,i,fiii,/s  iif'oii  111,  malcrtiil  itiiit 
(oHiiitioii  of  til,-  pipi . 

l"or  example,  the  resistance  to  flow  is  nuii  h  less  in  a  ^;las^ 
than  in   .111   inm   pipe,  and  is  ,if/>r,>xnnittf/v  tu-iir  as  ^vn.w   m 


H 


DARCYS   /:A7V:A7.V//;.vr.//    KliSLLTS. 


'■•7 


pipes  whiili  li.ivi-  lu'C'iinc  iiicni'^ircl  with  ii-^o  .(■^  in  iii'w  cUmii 
pipe--.  It  niu^t  111'  ninriiilHircl.  Iihwcmt,  that  .litiiuu'^h  iiunu'i- 
<)iisc\])ciinK'iit--  li;ivc  hocii  m.ulc  witli  new  pipc^,  there  h.i\  e  l)een 
comparative!)-  tew  (.'xperinients  uitli  old  pipes.  Thin,  in  pi])es 
in  which  the  veincitv   nfflow  exceeds  4  ins.  per  second,   Darcy 

■ ,     1  ■  /    A  ■ 

ConsKlereil  it  iiiitl  tuirect  ti<  express  </  I  :i=  -   l 


Ml  th'-  ti>rni 


f 
g 


fl 


d  i)i'in;4  iht.'  (hanii-ler  "t  tlu  pipr,  ami  <<  anil  li  coefficient-i  to  In 
•  letermined  h)  e\])erinieni  I  he  lollow  in;,;  \aliie-.  fur  it  and  /> 
are  L;i\i-n  hy  1  )arc\' : 

•<  ti 

V  "\    d;a\\n    u  riMi.L,'ht-ir<Mi    ii|-    siniMiih 

cast-iron  pipes.  "01)1345      .00001 2(>73 

I'"'ir    pipes    with    snrl.ue    iii\(ie<i    li\ 

li^ht  incrustations.  ooo^O'j;      .oooo:?!;oS 

\\  iiltont   --ensiblN'   .dterins^   the  \ahies  of  tlic-'<    ^   lelticients 
t'jrv  >  an  l)e  put  into  thi'  fciih)svini;  simple  Inrm: 


g\       1 2d' 


d  bem};  the  dianieter  in  feet,  and  k  heinj,'  .005  nr  .01  accor<linj,' 
as  the  pipes  are  clean  or  have  become  -^li^^litly  incrusted. 

(/')  / /ir  tofffuuHf  h  is  iii'f  <i"is/<i>tf,  hut  'iiirits  s/i!;//(/r  l<oth 
with  the  iitttiHitir  and  tilt  viloiity,  its  roliir  Uiini»is/i;iii^  iis  «/ 
or  r  iitirmsis. 

In  jHactici-  it  is  assumed  tliat  /'  is  convt.int  and  the  error 
involved  has  the  ailvantaj,'e  of  nivin^;  to  the  pipe  a  lar^'cr  sec- 
tional area  tlian  is  act'-all)'  re  (uircd    for   a   j;i\iMi   dischar}.;e 
Thus  ailowame    i-^  partially  ii>ide  fur  the  incrustations   uiih 
vinch  the  vurf.ue  ;jraduallv  beconcs  covered. 


I2S 


ll-IRCYS   h.<riJ<l\U-S  I   U    KISIITS. 


iJ.iicy  [Ml  i[)()-,cil    Id    inr'udi'   all    i.,i-,r^    iii    tlu'    iiii>ri-    ;_;cnL' 


luiin 


in  \\liiili,   fur  lu.'u   .iiul  ^iiUMitli  iicin  iii|K's, 


a  -  -  .0000 1  350 

If'    =      (  ;(  iri(  I?  If)^  ; 


li  —  .000012402 

fi'  =:   ( a  K  i(  II « )  1 1'}  1  sri 


1  lii^  v.iliii;  for        i>  r.MiU'  it  (Air  ii^cd. 


TAHI.l-:   (UVlNi;    DARCYS   NALTI-.S   OK  /  I- OK    Vl-I  <  n  !  ri  l-.S 

i:.\(r.i;!)iN(;  .»  in.  pkr  skco.sd. 


Iliun  \'.>  ur  .  I    •  11,1,11  V.lil.'  ..(    '.  Iinni  V.ilur  Ml    < 

„(  .1  "I 

I', p.-  I'll"  I'il'l- 

Ml  N' w         lfi<  ru>.ic.l  111  Nrw         IlH  nislrd  ■"  Nrw         liuMisut 

lii.hri  I'lprs  1'ii.ti.        Imh's.  Htpv,.     ;     I'ipts.     '   liulics        l'ip<;»  I'n-.-.. 


3  .oo(>'7      .i'i.».^.1 

4  I   .(lof)];      .ni3j 
<;  .iKKo        .011 

(1  .oo|!S3      .01 167 

7  ,    .KJJ7I  ,   .0114; 

5  -tosfaS  i   .otiaj 


i-5(l 

III  1  1  1 

*»- 

IHlt  II) 

.01037 

'. :! 

.i>'>?43 

.IIIIIS3 

3" 

.00517 

.01033 

I« 

.<lo^^,^ 

.oi<rfi7   . 

v> 

1  ion  1 4 

.oiojH 

IS 

.i»)528 

.■>I05(> 

41 

l>0?12 

.01034 

21 

.l>ilH24 

.01043 

4-* 

IKI5I0 

onui 

:\ 

..,  ,c_.  I 

'ilOp 

'1 

IK)50() 

.OlOli) 

Wii^h.tili  L'lM'stlu   tiiriniil.i 


II I !  ( I  — 


.(.10421) 

S  :■ 


I'nisciiilli's  evpirinuMUs  imlicjitc  tliat  tlic  surface  friction  in 
capillar)  tubes  is  tlircctly  priiportional  ti)  tlu-  velocity,  !)iit  in 
pi|>es,  in  ordinar>-  practice,  the  iVictioiial  resistance  is  certaini)' 
more  iiearl\  pri>(K>rtii>nal  tn  the  siiuare  of  the  velocity,  anil 
must  he  largely  <iuc  to  edilies  which  .ire  the  more  reailily 
fornieil    .is    the    viscosit\    iliiniiiislics.      This    viscosity,    aj^.iin, 


POISF.UILUrS   E\PI  KIMIMS. 


1  .'9 


inci"ca--i-^  .1-1  llic  U'liipi  r.itinc  lall-.,  .mil  the  -^iirfiuo  fn'itii)ii  is 
(liiiiiiiishcd  b\- ahoiit  i  \nr  >  out  I'M- (A  tr)-  li-r  of  ;  1-.  iii  the 
lcm])<.r,ituic.  riu-  rc-i>t,iim-  h>  t!u  motion  ><\  ,i  hodx  in  w.itir, 
or  to  the  llow  o|  w.itiT  .ilon^  a  siirfaci-,  i--  cvidcntl)-  of'  two 
kind-',  tlic  one  diif  to  surlaci-  >.nnt:ict.  tlic  otlu-r  to  the  torina- 
tinii  ol  nldic-;.  Ilclr  Sliaw  \  ixpcrinuiits  (.k-arU  -,li  ,\\  the 
cttccl  ot  -iurlai.  <  lontact  upon  Ntrcain-iini-  motion  ,ind  the 
manner  in  whicli  the  motion  i-  modilied  h\-  il-.e  pieNence  of 
obstacles  n"ran'-.  Na\al  Archileel^,  i  .S<j7  ijS),  while  the  two 
kinds  of  resistance  an-  ulainh-  dt  inonstrated  1)\-  the  interestin' 
experiment',   ot   (  )sl)oi  ne  Ke;.llold■^.       1  he  water  llow  -  throu!;!i 


+  - 


i    ■  -, : :.  ■  -rfj 


In;.   10. 

a  ;,das>  pipe  ./A'  li,i\ni-  a  Iruinpel-shaped  niouiji  .;.  A  ^-jass 
tube  r/'with  a  funnel  /.  ternn'nate>-  in  .i  jiipette  /,  the  .ixis 
ot'  tlie  pipette  In  Hii;  in  line  with  the  ,isi>  ot  t!u-  pip(  Tin  tnhc 
i^  tilled  with  an  anilin<-  dye  which  i-.  allowed  to  est  .ipi-  tluou-h 
the  pijjette  in  a  thin  thre.id-like  stream,  the  dischari;e  l><  in;.; 
^'overned  b)-  a  ^ni.dl  ;  o.  1.  So  loiij;  as  the  velocit)"  of  flow  \\\ 
the  pipe  does  not  exceed  a  certain  \,due.  whii  li  K<\iio|ds  c.dls 
the  ,/•///< (//\elocit\-,  the  aniline  threail  i->  unbroken,  ^ii  that  the 
motion  of  the  water  is  undisturbed  a\\k\  must  be  in  parallel 
lines  .\s  soon  a-,  the  critic.d  velocity  is  e.xceeiied  the  colore<l 
thread  is  broken  up.  liecomin^  simiou-  it)  eliar.ufer,  and  tin 
jiarallel  stream-line  motion  i^  ioinplitel\-  destrovi'd  withni  a 
\er)'  short  distaiue  from  the  mouth  of  the  pipe. 

According,'  to  Ke\  nobis  the  critical  velocit)-  (r,  i,  in   metres 
|H;r  sec . ,  is^ivin  l)\    th.    lonnnl.i 


vd 


„P 


>3° 


/■//  //'    I  Kii.l  It  >S. 


II   llrlllL 
\\  lliK 


\'-'-7') 


'  I  fi  .r  i-.ii>ill:u\  t  111  u-  .111(1  ^_^{"r  "rdin.ii  \   pipe 


J-S 


/' 


-:     I 


.oj^V'/ 


.OOOJ2  1/-. 


I 


/  l)oin-  tlu-  tfin])rr,ilurr  in  iU-;_,'ri'(-^  c«'nti.L;railc. 

It    h:is   hcci)  -liDUii  1)>    11.    1.   Hartics.   D.Sc.  in  his  oniutI- 
niint-    Mil    till-   spixiru-   licat    ■  >!"  water,  tiiat,  if  water  he    heated 
while  llMwin-  thr.>ii^li  .i  tnhe  at  velocities  less  than  the  critical 
\el()eit\.    the    temiieratiire    ilistrilnitinn   in   the    column    is    imt 
nnifiirni.       If  the   heat   he  ai)i)lieil   eleitricallv,  hy    means   ,A   a 
wire  threaiied  throu-h  the  tl^w -tuhe.   the  ',     '   water  llows  ahmm 
the  wire.  lea\  in-  tile  wall--  of  tin   tiihe  .ilnm-t  eiitirel)-  uiiluated. 
ll    the    heat    he   applied   t>>  tin-    wall-   i>t'  the   tiihe.    the    colder 
water  passes  iliM-u^h  the  centre  <.!'  the    tiihc  anlieated,  Icavin-.; 
a  elo.ik  C.I  lint  water   .don-  tlu    sid,  >.      In  mitlur  case  is  theie 
.in\   tendencv  to  mix  as  lon^j  a>  stream-line  tlow  is  inaintaineil. 
\    niw    method   for  <leterniininLr   the   critical    \elocity  ot   .i 
lliiid,     h.i^ed      '11     the    .ihi.ve    experiments,     h.is    heell     recently 
worked  out  In    I  h-.    n.irne-  .iml  t Kker  in   the  Mcdill  hedr.uili. 
lahoratorv       In  thi-   method,  .i  -ensitive   merciir\-  thermometer 
is   pl.iccd   exacth    in    the   centre   of  a  column    of  water   as   it 
emeri^es  from   the   tiihe  under  ex.imin.ition.  with   the  hnih   ii:^t 
he\oiid  th>    end.       The  walN  <>i   the   tiiix-  are   m.iintainid  .it   ,i 
const. mt  temi)eratiire.  vli;^htl>   ahovi'  th.it  of  tlu-  w  .iter  llowin;.; 
throii^'ll,  l>iit  for  stre.im-line  ll'-u   the   temperature   indicated  hy 
the  thernuimeter  will  he  th.it  of  tlu-  w.iter  in  the  head  -iil>pl\  m.-; 
the  const, mt  How  .       I  he  .irrival  of  tin-  i  ritical  velocity,  at  wliich 
stream-line  flow  hei omes  ettdv  in-  .mil  sinuous,  i-,  at  once  shown 
hy  a  sudden  --m.ill  ini  rea-'C  in  the  re.ulint;  of  the  tlierniometer. 
and  is  (liu- to  the   inixtun    -if  the    ".iti-r-hlm   next    the  -uriace 
with   the   eolder  w.Ucr   llowinj;   tlirou>;h  the   hmly  of  the   pipe. 
The  iKiint  is  very  sh,irply  defined,  and  the  nu  thod  is  in  man\ 
cases   far  more  .ipplic.ihle  and  convt-nient  than  the  usual  color- 
band  test 


SHIP   RISISTASCE. 


I  ;i 


TIu  ixiKTiim-iit'^  now  in  pmc^ress  in  t!io  li\  dr.uilif  l.ihor.i- 
t(ir>-  In  Mamr-.  ,iiul  (  (.kn  arc  Ijciiv^  niadv.  bdth  by  tiio 
tlK-rnial  and  id'h  T-ijand  iihiIlkK,  under  the  n:>>st  laviirabli.' 
onditions  tor  --icurini;  tlu-  pcrtcctis-  stead)-  mnditions  nccc^- 
sar\-  for  niaintainin;4  stream-line  llou.  Ihc  result-,  so  t.ir 
obtained  sliou  that  the  effect  nt  tein|Hi  atiu  t-  i-  \er\  marked  m 
altiTiii;^  tile  iK.mt  of  nistabduy  d  l1.i\\,  an.l  ili.it  tin-  \ariatioa 
accords  at  least  approximately  with  tli.  f.innnl.i  (|iiote<l  by 
C)sl)orne  KexiK.Ids  an  !  taken  fr..ni  i'.iisiLiilles  evpermients. 
T  lie  eltect  . 'I  pressure  has  been  studied  oxer  a  limited  ranLje, 
and  It  ha-  been  sjioun  tli.it  water  llnwm-  nnd.r  a  hi-li  head 
has  Ljreater  stabilitx ,  which  mean-,  that  there  is  a  <lehnito 
increase  in  the  \tlocit>-  at  uhicli  stream-Iiin'  motion  breaks 
down.  Indeed,  tnider  tlu  jiresent  .irran-eiiients.  it  has  be.n 
l)nssil)le  to  maintain  stream-lim  motion  to  \,t\-  intich  !iiL;liei 
veiocitii-s  than  is  possible  in  i  xperimeiits  carriol  .uit  with  tile 
appat.itiis  ii-.rd  by  Reynolds. 

3.  Resistance  of  Ships.  I  he  motion  ot  a  ship  tliroui;]! 
water  causes  the  jirodiiction  of  waves  and  eddies,  ami  the  tot.d 
resistance  to  the  iiinveir,ent  of  a  <liip  is  made  up  o|  a  frii  tioii.ii 
resistance,  a  wavc-iiiakinj^  resistance,  anil  an  ediK -m-ikinj,' 
resistance.  .\lthou{.,di  there  is  no  theor)'  In  uiiicli  tiie  resist- 
ance at  a  -iven  sjjeed  of  a  ship  of  ilefmite  ilesi^'ii  can  be 
absolutel)- determined,  lrou<ie's  e\|>erinients  render  it  possible 
to  make  certain  inferences  and  turnish  some  usehil  data. 

According;  to  Iroude,  tlie  frictional  resistance  is  sensibly 
tlie  same  as  th.it  of  a  rcctanj,'ii!ar  surface  nioxiiiM  vvjth  the  s.nne 
speed,  of  the  same  leii-^th  as  the  sliip  in  the  direction  of  motion, 
anil  of  an  area  equal  to  tlie  iiiiiiiersetl  surface  of  the  sjiip. 
l"..\periments  sectn  to  indicate  that  as  tiie  speed  increases,  the 
frictional  resistance  of  wcll-desi^^ned  ships  with  clean  bottoms 
is  from  yo  to  fiti  jxr  cent  of  the  total  resistance,  and  tli.il  tli.- 
percent.i^'e  is  j^reater  when  the  iiottoms  become  foul 

I  he  wave-inakni},'  resistance  is  cHiK-cially  affected  b\  the 
form   and    proportions   of   the    sJiip,    depemlin^',    for   a    j^iveii 


J.?2 


.\fiir  Ris/sr.-ix'.r. 


length,  upon  liir  proportions  dI  the  rntraiuc,  miildli-  body,  .iiiil 
niii.  I'or  cvfi\-  slii])  tlicrc  i--  :i  iiiiiit  ol'spocit  below  wliuh  tlic 
resist. UKc  i-.  .ippro\ini;itcly  ])ioporii(i!ial  to  tin-  ^qiuire  of  tin; 
■•pcc'il.  boiii;^  iliictly  liuc  t'  tVictioii.  ami  luxoiid  wlii.li  it 
iiKri\i>L-<  more  iapiill\-  than  as  the  s()Liare. 

Tile  eild)--rcsi>taiice  in  the  case  ot"  Moll-tonned  ship-,  .li.iuhl 
not  exeeeil  about  \o  pei  i^ent  of  tlu'  total  resistance,  and  i^  often 
inii.i)  less. 

I"'riiu<le's  ].i\\   of  resistance  nia\'  be  enunciated  as  follows; 

Let  /,  .  /.  he  the  lengths  of  a  ship  and  its  n;oi|i  I 

Let  J,  ,   J.,  be  tile  displaceniiMits  of  a  ->lup  and  it'-  model. 

Let  A",  .  A',  be  the  resistances  ot  a  ship  and  il-  model  at  the 
speeds  :^  and  .'•.. 

Then,  if 

:■  __  /,i  _  J,i 

the  resistances  are  in  the  ratio  of 


A 


Hence.  toD,  the  II   1'..  .mil  therefore  also  the  co.il  consumption 
jier  Ijour.  is  propf>rtional  to  A'.*',  that  is,  to 

J'      or     A-      or 


.iiid  tlie  loal  consumption  ])ev  'vi''-  i-  pro])oition.d  to 

J      or      /        oi       .•■". 

.A^jain.  A'  is  jtrojxirtion.d  t<i  /  ; 
that  Is,  t(i  /  X  /"'; 

th.it  Is.  to  r»  X  ^1: 

.md    it    Is    solUetlliH  s    lonviiiuiit    to    .-xpress    t|ic     ii'-ist.une    in 
[xjunds  in  the  form 


/'//'/;  I LOH^  ASSLMrilOSS. 


•:■  Wmj[  tlir  -iH-cil   ill  kiicit^.   J  tlii'  displaL-ciiKMit  in  ttins.  and  /[• 
a  OicllKieiit  (!r|i(.iKimi^  (ipMU  the  ty])i'  <>l  A\\\^  ami  \ar\-inLj  trom 
55  ti>  .S5  wlun  ilic  hottdin  i>  clean. 

i;x.  il  till-  Niu  V.iik,  uitli  a  ilispl.iciMiliiit  >A  lo.ixxj  ti.ns  and  i<(|Ui:- 
iii;^  20  cx):)  1  1.  1'.  tur  a  spet-d  oi  20  kiiiits.  is  takfn  a-,  the  model  for  a  ruv^ 
stcaiiiir  wIjilIi  is  Id  liuv'C  u  sptrd  i>!  Ji  kn>'~    tivn 

•     .•     ,  /-I    ' 

new  sU'aiiH'r  s  (Iisplaix-ineiit  —  Kj.oiaj!  -  1  ;,4<xj.  ai^pr'ixmKitclv. 


II.  r 


lU'W  NtraniiT 


\  20 


cxjo,  aiiprii.\mia.^i\. 


4.  Pipe-flow  Assumptions.  In  the  oniinary  tiicory  <il  tin.- 
Il(.\v  ■  ■!  wall  r  HI  a  pipe  it  is  assunu'd  that  tile  water  consists  of 
thin  plane  l,i>ers  pLi  peiuHcular  t^  the  a\i-  of  the  pilK,  that 
each  laser  is  ilrurii  throuL;h  the  pi|)i-  hy  the  action  of  ,i;ra\ily 
ana!  !)\-  the  difference  nf  jiressnre  on  its  plane  faces,  and  tliat 
'ill  iiipihl  iiKileciiK  --  111  ,in\  l.i\  c  T  .it  an\-  l;u  III  nioinent  u  ill  als.  T 
he  tciund  in  .1  plane  !a_\er  alter  any  intei\al  .  i|  tmu  ,  Ii;  '■ucli 
motion  the  intern.il  work  doiu-  in  detoninni;'  a  la\<  r  ma)'  l)c 
i^a-nei-all)-  disre_i;anled. 

It  is  turther  assniiii  d  that  there  is  no  cariation  of  \fli)city 
o\  11  the  ^iirlace  of  a  layer,  ami  this  is  eipii^aleiit  to  sax  iiij,'  that 
t.uh  lic|iiid  molecule  in  a  iross-scction  has  the  same  mean 
\  il(  11  ity . 

'The  dis.iort:cment  ■  i|  i1k-.i>  assumption^  with  the  its^lts  ot 
rei  lilt  cxpi  1  inii  nt.il  ii--.aii.lus  will  he  rek'rred  to  m  a  snhsc- 
ipient   aitu  h 

5.  Steady  Motion  in  a  Pipe  of  Uniform  Section. — Sine- 
till'  motion  is  to  he  sttad\.  the  s.iine  volume  (_' cu.  ft.  of  water 
will  .d\\a\s  arrive  at  ai)\  ^dven  cross-section  of  .1  sq.  ft    with 
tile  same  nuan  \elocit_\'  ;■  ft.  jh-T  second       Tlun 

('        .1, 

Jiut  siiii-i-  llu-  pi)>e  I--  ol  ion--t,inl  di,inii'lt.i .  ,/  is  const. ml.  ami 
lience  also  ;  is  constant,  so  that  tlie  mean  velocity  is  the  san»c 
tlirou^dioiit  the  ulmK   h  iiLitli  of  tin    pipe. 


134 


i/oH-  i\  rirr  n/-  rxiroRM  sm/ox. 


Vu:.   't. 


C(.ii-^i'-1lt  .-ui  rKMnrnt,ii->'  m.i>->  'A   the  llni.l  .1.1/:/:.  bounded 
h\-   the   jiipf   ami    by  tin    twt.  eri)ss-scctinn>  .  I .  / ,   /:/!.       Lot;// 
be   tlie    Uii-th    .  I  /:   "(    the   ele- 
ment,   tile    len-th    /   ft.    of  the 
]iipe  beiii;,;    niea'-ureil    almv.;  the 
axis  from  an\'  <iriL;in   '  '■ 

Let  -,  .-  -■  '/;  be  the  eleva- 
tions in  fei-t  above  a  datum  line 
of  the  centres  (.r  i)re-siire  m  the 
iross-seetions  .1.1.  /.'/.',  lespee- 

tively. 

Let/,  "  T"  i/f  bi'  tlu-  inti'Usi- 
ties  of  the  pressures  ..n  these 
cross-sections  in  jxiunds  per  siju are  lout. 

Let  /'  be  tile  perimeter  ot  the  jiip. 

Let  re   be   tile   speeiik    Mei.nliL   ol'  tin   watv  r   m    -pouiu'.s   pv  r 

eubie  lout. 

]\\:r/-  /h»/,-  /'I  '-r.i,  in.'  in  one  second  ,-.7J  lbs.  of  u  ater 
arc  trausfcrre.i  from  .  L  /  to  /:/!,  lallin-  tlirou-b  a  vertical  dis- 
tance of,/.-  ft.        Ihus  tlu    vvork  d,.n.'  by  -ravitv    per  second 

a  positne  ,iuant\t>   if./:  i-^  nc^'ative.  and  rtW  rrrv^r 

ll\,r/i-  Ihm,-  l<\  /'/v.s.vrt/r.— ■|"liit"t''l  pressure  on  A. I  i)aral- 
lel  to  tlic  axis  -^  /■./  ;   tlic  tot.d  pressure  on  /.'A'  i>arallel  to  tli^- 

axis     -  1  />  -•   '//•)•  L 

Therefore  the  total  resultant  prossme  parallel  to  the  axis 
in  the  .lirection  .^f  motion  -  .  /  •  '//■•  -'nd  the  uork  done   per 

second  on  the  volume  0  by  this  pressure  -^  -  C'    '^f- 

Noll..-Tii.r  work  dene- by  the  (.les-ur.- at   tlu-  pipe  surface  Is  nil.  as 
it«  dir.Tti.in  IS  ill  rinht  an^los  t,.  the  Inu-  ul  luotiun. 

K  ;.;•/•  Af>sor/>.,i  I'V  I'viitiomil  A"<  .v/.vA»//,<-.— I'nmi  the  laws 
of  tluid  friction  this  work  per  second  is  evidently 

-^    /• .  ,//  .  /V')  X  .-  =  -  J    ■  C' .  /■'  ••  '  •  'i^' 


IIOH-  IS    I'lt'fi    Ol     I  V//()AM;   .sl.CIKlW 


',;5 


the  sicjn  bciiif;    nc^jativc  a^   tlu'  wdrk   i'^   doiu-  ai^ainst  a   rcsi-^t- 
anci.'. 

Siiuc  tlu:  inntm)!  is  ■^tt  ad}  .  tlu  wdik  li'inc  1)\'  the  cxtirii.il 
forces  nui-^t  l>c  iquuakiit  t<i  tlu-  w  > 'ik  at)--<irlii  d  hy  tlu  iriLti"iial 
c-^istmce,   aiul  luncc 

_  :,(J  .  ,/;  _  (J  .,//>—       (J  .  /■<  :■> .  dl  =  O, 


or 


,/:+'+,.  .  '//  =  o. 


IiitLiji-atiiu 


-  ^  i-  -|-  — -  .  /  =  a  constant  =  //. 

Ik'  .1  7i ' 

SO  that  //  It. -Ills    per  ])<innil  'iftliiul  i^  tiic  imili>i-ml\   ili->ti  iijuted 
total  cnimtaiit  ciut;^)'. 

",  is  calleil  tlu    iiyclraiilu  imaii    iaiiii;s  of  a  pipe  .iiui  will  1)C 

■IciU'ted  b)-  III. 
lake 


the  \aliu'   .idiiptcd  in    oidinaiA'  i.rattiee.  /  beiiiL;    the  eoellu  Kiit 
ot  fnetion.       1  hen 

p         fl   V 


w    '    in  2g 


H. 


l.etc,  ,  A^.p  be  the  ele\atioll  abo\  e  datum,  the  area  of" 
tile  cross-sectiii!!.  atid  tlu'  itUe!isit_\  of  liie 
])ressiire  at  an\  pMint  \  nn  the  axis  of 
tlu-  pipe  di-^tant  /,  from  the  orii,Mii  [Vv^- 
SJ). 

Lctr,,  ^-i,,./".  be  the  ci(.-\atioii  al)in.e  d.itiini,  the  area  ^t 
tlu  I.  ross-seetioii.  uid  tlu  iiiteii--ity  "\  the 
pressure  .it  an\' otlu  v  piniit  )  mi  tlu  avis 
<ii--t,mt  /  ffoii!  the  (>riL;iii  (bu;.   '^^1,. 


13''  //()»■   /\    />//•/     (>/     /■V//(WVf    SHLTHM. 

Then,   triiiu  ilu'  (.-(jiLition   jii^t  ilcdiurd, 


/•,  ,  yA  •■' 


A  ,  /^j  •■' 


1  ic'iicc 


=,+  ^:)-(.+{=^  =  '" 


Ji'  /  w 


(/.  —  /,  I  = 


I.  t)i  iiil;   tlic    U'iiL;th  /,       /,  <il   llu-   ihjii-  Ir-ixmiu  tin    two  ])Miiits 
.\'  an,l    r. 


Fic;.   '2. 

I.(t  \(Tti\.i!  tulx  I  ])rcssiir(.'-c<>lunins  he  iiisortc<l  in  the 
yiipi'  .It  A  .111(1  at  r.  i'lic  water  «ill  rise  in  tliese  tubes  to  tile 
IfveK  (    ,inil  /*,  ami  c  \  idiiitl)' 

A       •-•••^'•V-^  A. 
A-  ^' ■/'>'+ A. 


/,,  l>t'iii^^  tin;  intensity  ■•ttlie  .itnio^plieric  pressure. 


hU)ii    i.\  I'irt  Of  i.v/AOA'.v;  .stLl'iox. 


';>7 


i  Iciict  ,   if  CV  and    /M'   ,irc    produced    to   inert    the   datum 
line  III  /;  and  /■, 


ami 


'lluri  liuo 


«-.,  +  ^'    =  r:.,  +  />)--f    -^'^    =    /V'-f    '^". 


(=.  +  &) 


A- 


c  /;  -  />/■  -  /^c;  = 


W    2..'- 


<'/'  hcinu;  the  [joint  in  wliu  h  the  hnri/oiital  thrmivdi  c  meets  /  /- 
proihiced. 

/)(,'  is  i-.tUeil  the  '■  \iitna!  tall  "  i>t'  the  pipe,  beili^'  the  tail 
<il  Ie\el  in  tile  pressiiic-edlumiis  ;  and  -^inee  th.  re  would  he  iu> 
fall  ol'le\(l  if"  the  fViition  were  ml,  / >( ,  is  said  to  he  tlic  head 
\'<^{  m  tVietiiin  in  the  (intaiue    \  )  . 

Denote  this  head  hy  // ;   llun 


//=. 


ni    2^r 


ani!  therefore 


This    ratio  is    designated    the    virtual    slope    oi"  the    pi]ie. 

and    is  the  head  lost   in  friLtioii    ]ier  unit  of"  lenL,'tli,       It  will    he 
(len<  ited  hy  ,',   -o  that 

h       .         f    V 
L      ^  ~   m  2g 

ll  llie   sc'.tioii    ot'  the    pipe    1     a   eirele   ot   diameter   </,  or  a 
S([u.iie  with  .1  side  o'  le!i;,;th  ./,   then 

./  ,/ 

'"  =  r  ^  4  • 


and 


iNi-LUi:\a:  oi-  rin.w  y.st././.v.-i/vo.v. 


f 


\vlifrc    <r  =   "     ami  /■  i>  tlic  r.uliiis. 
A 

6    Influence  of  the  Pipe's  Position  and  Inclination  on  the 

Flow. 111    li-.    SJ     l"lll    (    /'.        Ni>\^    ■'llUr    tiu-    l.dl    i>r    k-\cl   i//) 

is  i)r<)]iorti<inal  ti>  /.  tin  Iitc  -^iirfi.ci'  in  :my  dtluT  lokmin 
between  X  and  )■  nui-t  .il-i>  1h-  (.n  the  liiu  (  /',  riiii-  the 
nressurc  /■'   at   an\     intennechatc    imint    M  ch-taiit    i   (=.\.]/j 

re  a'  '-  •' 

1 1  -nee.  at  e\er\-  point  of  ,i  pipi'  laid  behiw  (  /',  the  lliiid 
])rcssurc    ( />       exceed-    tlu'    atiinispheric    pre-^-urc    i /„  ■    t)>-    an 


.iinomit  a'  .  a/ .  \  ,   -n  m.u    n    i"  ■■> 


111.'    in     CI  ti  ■  li     I     utt  w      the 
■ \  '  \.   ~  -     


water  will  tlow  nut  and  there  \m1!  he  m.  tendeiie\  on  the  part 
of  tllc  air  to  tlc-iw  in  In  pipe-  -"  pKued  \ertual  bends  may 
be  introduccii,  care  bein-  taki  n  ti.  i)rovide  lor  the  reni(i\al  (jf 
tile  air  which  nia\-  eo'h-et  in  tin-  upjier  part-  <it  the  b<-nds. 

It  the  line  of  the  pipe  (.oiiuides  with  (  l\  \  e  ,  with  the 
N'irtual  slope  or  line  of  tree  -urtaei  K'vel,  M\  -  o,  and  the 
tluid  prossvirc  is  e(iii,il  to  that  of  the  atrnM-,plure,  If  holes  are 
1U)W  made  m  the  pipe,  it  ean  ea-ily  be  shown  In'  experiment 
titat  theri  will  be  neither  any  tendenc}/  on  the  j)art  ot  the  water 
to  (low   out  nor  on  thi    part  of  tin    air  to  tl,,w   ni 

/I 

Next  take  (  (  '    --   /'/''  —  '  '\  and  join   (      /'  . 

d' 

Itthe  pipe  is  placed  ia  an\  po-itioii  between  t.  /'  anrl  CD', 
.l/.\' becomes  ne;^Mti\c,  ami  tin  tliiid  pre. sure  in  the  iupe  is  less 
than  that  ot  tilc  atnio-phere  It  hoK-,  ,ire  made  m  tin.  pipe, 
tiiere  will  be  no  teildencv  on  the  part  ot  the  wat.i  to  i1...'a-  out, 
but    the    air  will    tlow    in.        I  hu-,   il'  .i    piju-  ri-e-    abo\e  th.-  line 


I'lrr.  n)RMt  i..i.. 


^39 


t)f\iniial  •-loi)f.  tliLir  1-  .1  (laiv^cr  iif  ;iiv  iKcuimilatinL;  in  tlie 
pipe  .tii'l  imimliiiL;,  >'\  ]>crhA\)<  \\lii>li\  -ti.ppmL;.  the  tlnw. 
Nil  wrticil  1)1-11(1^  sIkuiM  he  mtrtKluccd,  ,i-  tin.'  ,iii-  i-^  c.isil)-  set 
fr(-c  ami  would  cnilcLt  in  tin-  -.iiiiicr  i)art'-  "I'  tin-  hciuis,  with 
the  fftVct  <if  impeding-  tin-  ili>\\  and  ot'  acting;  drtrinn-ntally 
iiimn  tiic  water  itself,  which  the  lihcration  "f  the  air  renders 
less  w!l.)lcsome.  If  the  iitle  ofpijje  ediiuiiie-  with  (  / >' .  then 
the  tliii.l  jiressiire  is  nil. 

l-ina!ly.  it"  the  pipe  .u  an_\-  pmiit  li^e-  ah<i\e  (  /' .  the 
pre->iai'  heconics  neu,Mti\e.  w  hiell  i>  inipn^sihle.  In  laet,  the 
e(MUiiuiit\-  ot'  rtou  i-.  desir(i_  ed.  and  the  pi])e  will  no  longer 
run  full  bore.  .\iv  will  he  disenL;a,L;ed  and  will  ri-e  ,iiid  eollei  t 
at  the  point  in  (iiiestioii.  ^-o  that  in  order  to  prevent  the  llow 
hfini;  whollv  inilH-ded.  it  will  he  necess,ir\-  to  introduce  an  air- 
thamher  .it  this  jioint  from  w  hii  li  the  .ur  t  ,ili  he  reiiii  i\  ed  w  hen 
re(|uired. 

Ni'ii.  in  tliL-  prccpiiiiiis'  i'  tias  lu'cii  assumed  liiat  the  ipinr  is 
str.iiylit.  if  tin-  pipe  is  curved,  so  also  is  tiie  iiiic  <i{  vii  iual  .siopi-.  lu 
ordiiiarv  lir.iClitv,  iioA.vor.  tlic  v<riii.il  (  iianges  ol  level  in  a  pipe  at 
flilTiTcnt  points  arc  small  ;is  ennip.ired  with  the  lent;tli  of  the  pijio.  and 
dislances  measured  alon^-  tiip  pipe  are  sensibly  proponioiial  to  distaiues 
measured  afnj,'  the  liori/ontal  projection  ol  the  pipe.  Hence  tiie  line 
t.f  virtu. il  slope  niav  be  assumed  to  he  a  straiglil  hue  williuul  error 
of  practicil  .mportance. 

7.   Formula?  of  Darcy,  Hagen,  Thrupp,  Reynolds,  etc. — 

l)aic\  ,irr,inL;ed  tin  re  iilt--  of  lii-  expel  iiiieiit^  111  .1  tahle  dr.uwi 
up  a-  |i  I'.low  s  : 


Vei«>(  itj'-;  in  in 


I>iainetcr. 


to  3  m./iec. 


/) 


no 


riri   loKMi  1,1-:. 


I 


I  lu-  tir-^t  I  nliinin  i,M'v<.''i  tlic  ^f\  (.ral  ili;iiiictir';. 

i  he  sct'iiiil  idliiinn  ;>^i\<'->  tl;c  tnm  x]iiiiif!jn;_r  ^cttioti.il  .irci^. 

I  he  rcm.M'niit;'  whimii-.  :.,i\<-  tlu  si\(i,il  \-clin"iti(-~  ■•!  \]i>\\ 
fiiim,|in-^.  I  I  111  i  ii|i  t.i  lo  tt.  (^  111.  )  jK-'  "-rtcU'l.  ,mi!  i  ,u  h 
\  tli  H  il\  1 1  'hill  111  i--  -.nlidix  iilfil  in '.I  I  t\\  (1  I  dill  I  nil--,  tlu-  "Uv  i;i\iiii,' 

tin-  I11--S  of   Ik, 111  I    pel    niiit    n',    iciiLMii,   .iiid  tin    ntlici     i.uilli,' 

till-  <li^i.li,iii;o  {0\. 

All  c\;iiniii;itinii  (if  tli<  t.ihlc  <>(  |)air\'>  rt";\ilt-.  slio\\>.  tliat 
itpf'i  .xiniiitJy  t!)>-  Insx  !i  w  (lii<Htl\  pii  ■]  loiticn.il  td  tlic  li'iij^jth 
/,  di  |ii|H'  iiinli-i  1 1  iii--ulrr.itidii  aiul  td  the  -.([inic  dt  the  vt'li)city, 
7',  and  i--  iimi'^rly  prdpLitidiial  td  tlu    dianntir  il. 

'lilt  rrtdiT 

T  .t\       ..' 

n        ,  V  ,,  L    , 


\K  lu 


f  .7  I        N 

a  =       =  _   1  -I-  •  n    I  ■'■'  "i 

g      %\        ijd/        '  •      '-' 


In  1  la^;!"!!-^  tdiiuula.  \  i/., 

h       av 

L        d    ' 

till-  \  aim  •>  dt  a.  ;/.  and  1  \  ai  \  w  itii  tin-  ,  i  Ini  it\-,  tlic  diaiiu  ti  r, 
and  null    the    roii^hni'ss  dt    tin     -uilaii  1  he  ,.     iilt-^  dhtaiiu-il 

h\  this  foriinila  aid  in  .Kidnl  uitli  t!ic  losults  dt'  I'r.n-all's 
c.\|HTiitK'nt>*  \\i''i  ))i|)i.s  in  ^ond  idnditidn  ,\\\i\  dt  dianu  trrs 
varying,'  froin  o  ft.  tn  4  ft.,  uhcn 


'1 


1   N 


iiiil       I         I    (, 


})tit   ilu"  a^jrccnH-t\t  is   nut  su  i.lii->c  if  tlic   jiipi'     iirf;>cc  is  \crv 

Sllldlltll 

If  thr  ])i|i>'^  ail'  1 h  ,   till'  .11  ipi  1  >•  liii  iti    i   MUr--  dt   tllr  itlilu  C-. 


are 


a       .cxK^r,     «  = 


ami     .r  =  1 . 1 , 


PII'F  lOKMll  .i  .  1  ;  I 

hut    tliLSi.'    wihiis    iiuHt    iiCLC^s.inly    \  .iiy    with     i\ii\    iIiIVitj  nt 
class  (i|  i>iiir 

\,inoii-  lUoilituMtinii-,  III  II,i;4otrs  foinnili  li,i\i-  linn  pro- 
posctl,  .111(1  |)L-rliu|)-i  ^>n^■  dt  t!u  ln'-t  i-  ili.it  (.(irit.iiiii  <l  m  ,i  [i.i|ut 
by  Tlirupp,  read  hcforc  tlu-  Socioty  of  I'.iv^inoci -.  iI.Miulon)  in 
I  S1S7.      It  ni.i\   1)1-  w  rittni 

'  r    I  I  ''  '"'   '' 

.  =  cosc-c.  Ill  sliijic-  aiif;lc  1^    .    u^  J   , 


X  -i-  y 


y 


Ix'Iml;  Mih-lituti'd  Iim     i   w  Iuii   ;//  i-  small  I  In- 


v.iliH-,   (it    ;/,  ,  ,    1,   I,  .mil    :.    toi'  a    ;>ipi-    "i    ili.imirl.    .iri'    L;i\cii 

b\-   tllr   t'lll.iW  III"    t.iMi- 


.r  l.itr. 

Wrounht-irnn  pipf* 
Rivfifil  shoe!  irmi  pipes 

N'l  «■   '   .1^1    ll.'II    piprs 
I.l-.l    p,|...    , 

I'urc  ifiiiftit  riMnlcrini;  ., 


I    1  74 

'      I.IIS 


Brickwork  (smimihi 

(riMfh  .  _■  .,.. 

fnpl.ineil  plank.  .....  j  ixi 

Small  Kt.ivfl  III  tenu-iK    .  . .  j.iid 

larnc a.iMi 

Hanimri-lrrssed  muiitnry.  j.mi 

Earth  (im  vcuclation  ) j.oii 

RouKh  st..nv  c.irili    .  ].i«j 


o. 0047^7 
i>,i)i>5()74 
0.01H347 
().oo()7SJ 
0.005224 

(l.<MI4<)<)<l 

<i.iKi(i4ii{ 
o.iio774<i 
o.iH>8H4^ 
II  fN  1^451 
o.iii  I  Si 
11.0141:1 
0.01 1 17 
11.111 « v> 
11.1)3144 


O.'ls 

i>.h77 

o  f)7 

of.t 

o.hi 

o.<,7 

0.61 

ii.bi 

o.6n 

I). (lis 

o.0(> 

II.  705 

o.Mi 

"  7J 

0.7* 


u.olS 


0.07 


0.01234 
I1.03J41, 
0.03938 
0.07  540 
11.07835 


o.  5i> 
o.  ;o 
u.(m> 

I.IXI 

l.ui> 


()sl),irni'  K(  yiuilds  has  piiipoiiiiilr'i  .1  mi)1|>!i'  l.iw  nf  ri-sist- 
aiKc  1  nr  k  iiii,'  tlu-  results  ul  rniscnilli-  .mil  l»aii.y,  .iml  t.ikm^' 
into  ail. ..lilt  ilu  iHcHt-  ..I  viscosity.  tcm|K>r;iturr,  otc.  I'his 
law  may  bo  expressed  in  ili.  r.nti  mIh-  units  beiiij,'  .1  |M,,t  .md 
a  sccund) 


i       tlie  slop«» 


B         V- 
Al'     ■  d 


'4-' 

ill    wliiili 


/■//'/      /()A'\f/  ;..•?:. 


A  —   lAj\-  X    1".   B  -^    ;ii.,S,   and 


A->7,^h/  -;-  .(K>~iji  r/*. 


I  !)fin;4  till  ti  lupnatiirr  hi  ili-;^m--  i  i-iUiL;r,i<li',  .\[)i)r.  >.v- 
m  itrl\  ,  ilu-  iiulfx  V  i-  i  it  tlu'  iritic.il  vclniitx  is  not  c.\- 
cccdiil,  ami  I  ;  to  2  lor  xalui-.  mI' v  -rcatcr  than  tlir  critical 
vclucit)-.  Acciinlin-  t"  Inwin  tlic  iiidrv  i.f  d  i--  imt  cx.utly 
3  —  //ami  sliMulil  he  (111' riiiiih  il  iii.lrpi  iuUiul\'.  I-'ur  a  mii^'li 
siiriacc  //  -=  z.  Inr  a  ^iiUHitli  last-iron  |)i|ic  //  i.ij,  aiui  tor  .1 
lead  j)i|H'  // —  1.;^^  a  liiiiitatidii  wliicii  is  analo-^ous  to  that 
found  1(\    I"ri>ndi'  in  his  i'\|n'rnniMUs  upon  surtact-  Iriction 

II  iiii^  Ik-  noted  that  the  simi  ot'  the  i-\|ionents  ot  v  and  d 
is  constant  .\\\\\  einial  to  ;. 

III  a  paper  read  before  tile  l\o_\al  Society  ot'  Neu  S'aitli 
Wales.  i.S(,-.  knil)l>s  nu.'stij^ates  the  ettccts  ol  temperature 
and  record-  the  results  of  a  nunilier  of  e\]K,Tiiiic'its,  hut  tiie 
formula  he  deduces  is  too  complicated  to  he  of  inucli  i)ractical 
\  liue  and  reiiuires  t'urther  \erilii  ation. 

Fonrnie  lias  also  studieil  temperature  eltect  and  lias  su-- 
},'e~teii  ,1  tormnl a.  hut  his  results  are  not  complete  {Ann<ilis  i/f:, 
/'i')i/.\  1 1    i/i<iiis.\i,s,    I.S9.S). 

A'^ain.  a  simple  emjiirical  Liu  lonnictiiiL;  .■  ,  w,  and  / 
may  he  e.\|»ressed  in  the  lorin 


n'  \^  III'  h 


I  I  oi-dlicieilt  whose  value    is  t..  he   detertniiu  d  h)- 
Ti/  "'  6J.4J     •      /  /      ' 

3//         ■    till-  lorinul.i  m.i\   he  written 


c  \  pi  1  iiiiint  1  akinv 


thin,     if 


'•54    M 
II 

lor  \alucs  of  //  Iron)  .<x)S  to  ..nS  the  results  are  practicall\- 
tile  same  as  those  obtained  by  substitutin},'  the  same  values  for  ii 
in  Kutter's  more  complicated  |ormuIai(  hap    llli;  hut  while  the 


ril'i    iiiRMl  L.h. 


>4.? 


twii  tnrmiil.i    t  In^rly  .ii;ii'i--  in   urdmar}'  i.,isi-->,  tlic_\    bulh    l.iil  m 
(.■xtroiiic  L.iscs. 

I'll'  tiiiiniil.i  i--  .il--c)  i'ljiLilK'  .i|)iilii'al)li'  to  open  rli.niilciS 
(Cii,i|),  111  I,  ///  lie  iiiL,  tin  nil, 111  li_\  ill. mill  iU]itli,  hill  1  iittiiii 
li.!--  tniiiid  thai  will  11  ;//  In  -mall,  ami  I'-^iiciially  in  llir  <.a--r  1 1| 
oiirii  Mianiu  N,   ii  i-  pKtiralilr  \' >  u>i    tin    iikmIiiuiI  ixpriN^ii  m 


/'•54        ^\     '■■' 
—       I'"  '    • 


I  aniix    -  w  <  ll-knii\\  II  I'  ii  nulla  fur  ivmi.  |ii|i(-i  is 

wluh     I'"<isv  |,;i\(-.  |i.i   till     -anil    i.  a->i 

I  w  I  ///■'/'' . 

Ill    lS'17,    M     I  '\\    III    lii'^    I liioric  t/'iiu    Coiiraiit   l.iijuiiit. 
\\v    iiiiii-  111  111,4  a  niitii   iiii:i  stii'iid.    \\\\\:: 

1(11  iiiiK<  ,iisl-ii(>n  pifiS  :■  -    3f).4;;-/\i   -\rSr)\^\ 

••  ,iist-ni>ii  fif<<s  ill  sii:i,,     ,        20.  5; /-/(I  -|-3*'')!^- 
■|  n  tlnsi-  V'allot  addid  in  1  SSS  : 

imi  .uaiu.i iiist-iixii  /•//•'■»■      <"  ~-  S--5;''Ar  -f  ♦  /•);«. 

'Till    I  orri  •^I'lindini;  furniiila  .  with  w  fi<o,'  aiiii  second  w<  imits, 
;iri 

;   =  93. 24 j w/t  I    •    . r^oo  S  iii\\^\ 
V  =  52.51  ;////(  '         •      1-7  «  III-  -■ . 
r'  =  83. 24 ;«//'( I  +  .;«o9  Vy//);!. 

Vallot  .»i'<i>  nindirud  tin   (  \iiii--ii>M  f.  >r /i//<'.v  /'//  srrrin.  and 
(l(  iliiLi'ii 

7'  =64.788///'  /I  in  mctrii  unit  . 

c>r 

f  =  tjft.  2~ni'  I  ' .   .1  t(>iit  and     ri  •  uid  hem;'   the  nilit''. 


Ml       iiK.ti'iiic.-ii  Ki /'Kt:sF\r.iii()s  ni  I'lri-.  fORMii.-f. 
M.mniiv^.  in   iStjo,  l;.i\i'  tin-  fiirmiila 


1.4.S')     .    ; 


J/  hrini;  the  ^amc  as  in   Kiitti-i-  lnriniil.i.  Clia])     III. 
I-"lani.uit.  in   iSmj,  (icihn.til  t!i'-  >  sprcssimi 


ha 


'>! 


aii'l  l;.i\>    til'    lultiiw  in;^  \aliirs  |(ir 
I'nr  tin    pip< .     . 


!84 


•  •  UMll        ■     .  

••  ;:las.  ••  .   .         . 

"  u  rnti;;lit-iiiin  anil  .'.splialt.  1 1  jupr 

••  ncu   e.isl-iron  ami  lain<l  pipi       .  .  .  . 

•■  Itirlitt*-   in<'nisti(l    H"»it>    initio    111    <i*ryirt 


(     --   262 
,■  —   -»r»i-    4 


5 


8.  Graphical  Representation  of  the  /orraula  v       cm  i '  — 

rh>    prccrdiiiL;  funiiuia    arr    spcual  appluatKm-  ■>!  lli<    i;riKral 

in  wlihli    till     I.  "ittiiifnts  ,.    i    and    1    tur   an\'  ^riirs  i.t    rxpcri- 
mcnt>  can  l)f  ^rajilni  ally  diti  1  iiiiiui!  ni  tin    IdIIi  iw  m:;  ni  mm  t  : 
lakinj.;  !i)'_;aritlnns. 

lii;^'    .     =    Ik-;   ,       '■       I     In;.^    ill      ;      y   lo^  /; 

anil  it  /.    Is  a    p.irti*  iilar  \,ilii>    nt  ,•   ^  mn  spniulini;    tn    i  value   7' 
ot  .-  . 

lot;  .-J   -    loy;  ,    ■\     1    lu^  w;  -f   ,•  'njj  z^. 
1  hrn 

log  ;•  -.  Inj;   r',  -  )'  [U^V,  '  "   '"«  'i)- 

nnd  is  tlu'  i(iiiatinii  tn  .1  straij;lil  lim  .  tla    k  i  taiu;iil,ii    iimrdi 
nates  bcin^'   til-     In^.iritliiiis  ul    ,    and  "I   ,       Si  kctin^'   any  set 
of  experiments  ami  plotting  the  torrespondin^;   vmIims  of  lop  7' 


(jK-ti'iui.Ai   h'/:rKf:s/:\ r.iri(>\  ot-  rirt   kikmc/.i.       145 

and  iciL^  /  ^  ^(i-jo  i.t  p.ir.dlil  ■-ii-,ii;_,!it  lines,  nKiiiud  at  .1  inn- 
staiit  aiiL;U'  tan  '  r,  i--  oljtaitK'd  I'nr  all  tin-  \(lni.  itic-.  i.<iiti'- 
-]ii  indin.L;'  l"  \'>'^  /  r;  o  or  /  :-  l,  i.e..  at  tlu  mti  r--rrti'in->  nf 
tlu-si-  lines  witli  tlu-  avis  "('"loi^  ;.'  tin;  ^iiKrai  i\|irrssi(iii 
bcconus 


Tzz  I  in' . 


'1  .ikin''   Ii  iL'^.iritlnus  a>Min, 


1<)^  (     •    .1   liii^  ;;/, 


and  it        i-  tin-  \alin'  nf  .  •  1 1  .rn  s|)ii!idiii;4    ^"    ■'    p.irtuidir    \  aluc 


l'»k'  ■■■,  ^    '"J,'  '  -;   -'  1".^  "', 


1  lu  Tl  toIC 


a?i(l  is  tin-  eini.-itinn  tn  a  str.-ii'.;lit  liiu  .  tlic  n  i  tan  jil.u  •,  en  irdi  nates 
bein^  tlie  |ii;.,faiitlnii-  nl  ,    and  'it  m 

I'lottini;  the  dilli  lenl  \.ilnes  of  In..;  /,/  1 1  i|-ris|)oni!int^  to  the 
]iuticnlar  values  ot  lo^  ;•  in  (|uestion,  a  scrii--  "I  pai  dl'I 
strai'^lit  lines,  ineliiied  at  a  constant  an^le  t.m  '  .r,  is  obtained. 
W'iieii  loi|  ///  =  o,  or  III  1,  i.e.,  .it  the  intersections  of  these 
liiu  s  sutli  the  axis  of  "  h>Lj  ."  ,  the  u;enerai  i-spression 
beci  lines 


'I'hei  I  fore 


loU  ;•  =  lojj  <•, 


and  the  coeflRcicnt  r  can  be  at  onci-  obtained  from  the  <h'a^;rani, 
as  it  is  the  value  of  loi^j  .•  correspondin;^  to  /;/  -  1  aiul  /  i. 
In  i8()6,  !  iitton  completed  an  adnnrable  collaboratioti  of 
the  most  mijioit.int  sets  <»r  experiments  on  pipi-llow,  m.ire  tliaii 
1000  in  numbii,  and  varyin;.;  wuleh  in  diinutii  and  kind  of 
PUk:. 


I4tj 


//'/  I i(>n    ni  ii;ix.tMS. 


g.    Diagrams   Showing    Results    of    Experiments.  ~  I'.v 

iiu.m^    "I    tin      nuthiul     jii-,t     dt^i-iilud,      I  iiltnii     h,!--    jili.ttid 

cif  I  XJH  IIIIK  IUn    "11    |il]H-l1ii\\     ,1--    |i>ilM\\~. 


■ 

LOO 

' 

,,-'' 

> 

.  ^  -^    -^ 

.                       i              -       ,      - 

0 

o 

0 

-1 

• 

^ 

■      ^ 

1 

/ 

' 

1 

'  ^ 

.'f^^ 

WOODEN  PIPES. 

'  ' -^ 

^ 

■    \ 

.  . 

o  -^ 

LOOS,  or  «  AND  1 

I          1 

p 


.0     .!•     a.      .J     .1      .li     .H     2.     .■•'     .1      .li     .'I      I.  :i.     .1      .<>     .f<     0 

Fli..    .S3.    -I'luw  thriiii;;!)  W'noilt  n  I'iiif-. 

I     H.imilton  Smith.      Scrips  10 '/;          .o3')3 

.V    I  ).iii\  .111(1  Hitziii.          •'        5 J .    .    in          .303 

;.    1 1. Ill  V  .unl  M.i/iii.          '        51  •  '"          •  JOS 

4.   (  1.1  T  kf,  M.  lull  I  si. 1 11(1  (  0111  111  It ";        1.50 

4  siTios,  22  ex|K'rinu'iiti'. 

I'ormuht:  '•  -  1  2'y'«'*/". 


riri  I lOH    i)i.^(,HiM.s. 


'47 


■*.       .«        .8         3.       .5        .4        J        3         T.       Jt       A        .«        T" 

Kii;.   S4.  — Flow  through  (llass  I'ii>rs. 


D.ircy. 
Darry. 

Sniiili. 
Snulli. 
Sinilii. 


Scries  II.   . . 

•  MA  . 

•  9.... 


5  scries,  32  experiments. 
I-'ormiilii      V  —  141  to  169  »« '•'/■'■', 


iH  —  .04075 
m  —  .04133 
III    -:  .0191 
m  =  .01555 

HI    ~    .OIO4J 


MS 


nrt:-i-u  )h   Di.-iOK.-tMS. 


H.i. 


J.    ^.'    .1  ^  .'■.    .s    3.    .i    .!     .'■■    .X    i.    .-J    .1    .■;    .-    I.    .■-    J    .iJ    -s    0 

Fli;.  Ss.  —  Fr>w  tliromjh  N'W  Wrouglit-irnii  ainl  Asphalt-c  .lateil  I'ij.c^. 


I  D.my.      Series  i....  m  —  .oi 

2.  SllUlh.  "         'l.  .      ,    »/=r. 013117 

3.  Smith  ■'       5.  ■  ■     w--.(i2:j2? 
,(.  D.in  y  2-    ■  •  "'  =  .021^2 
S.  Smiiti.  .(....  w/  =  . 11211) 

(1  I). Ill  V  '         7-  •  ■  ■  "'  —  .Oliff 

7  Sill  all  1,2.  »;  — .02II) 

7.  Siiiitli  ■■       }.  .        Ill  -    .(,2I< 


If 


«.   • 


\    Klllll.UlM,        I 


III-    .114: 


■  (  ll.ihiiwalil  \ 

1).  D.ircv.      Sciitt  (. . . .  Ill  -.0324 

)  UlueRiilKi"  siphon  S  ' 

Ii.  C'Hiplel  . »;  — .1332 

II.  Ilirii.  Kesrtuss  Si.  .  .  »(-    .0837? 

1:.  li.irLV.      Series  ■<....»/- .o<)775 

It.  llien.  Si  til II' n  St i»  =  .  1247s 

13  IIjcii.      Series  5u..       ".=^.1247^ 

14  Khiii.ifui.  Stuttitarl..  /«  — .i(i?« 
15.  Uaroy.     Series 'i..      ";=.i(io7> 


\  Still  til.  ( 

'    (  N.irlh   Mliiiinilu-lil  \    '"" 
1 7.    Itien.  Moiin  pipe  ....    ■'. 
,     \  Siiiitti.  / 

''■     (    S'-ltll     lUonllllK-lil    *      "' 

I'i.    l).ui\.      .Scries  I.I  ..  .";.— 
\  Smith.  / 


21.  Smith.    lev. is  Creek,  m-- 

22.  L.tiiijie in  -~ 

2%.     lulilis.  Ko<  hesier.        «/  = 

23.  Ilien,  Stermhunzc  .  .  m  — 
J4,   Smith.  Iliimliii^  P'l'^  '"  = 

\  Smith.  I 

2";.  ■  ,-1       .  "i~' 

'     (  (  heriikee  pipe  \ 
■2(>.     Tlllihs,    Kocllester.  .  .    "/  - 

27.   Ciale 111  = 

sR.  Row  lain! ,  llinh  Meads  m~ 
2i).  "  "  "         W  = 

30.  "  "  "         lit  — 


2?  I 

23375 
307  e 

354 
34325 
50 
4167 

53^*5 

(.075 

75 
I. 'I 
020.S 

OJOs 
OJO'l 


3(1  series,  ii)«  experimenis. 

/■'.'iiiiu/r     I'm  asplialt-riMtC'd  7'  =  IJ'lto  iSS  ;«"-'/■". 

Fur  new  wru.i^lil-iron    v  ^   IJ7  to  165 /«''/■". 


rirHHOH     Dl.H.K.IMs. 


'19 


TUBERCUL^TED  OR  hL'=TLD  FIFE  Of 
RON  OR  LIGHT  MUD  DEPOSITS. 


4.    ..:     .4     .i;    .s    .'t.    :i    a     .u    .m    2.    ji    A    ,u    .f<    i.    .^'    .i    .ti    .h    o 
Fig.   86.  — 1-'1,.w    ilir<uiirli    TuLiti  iilated   or   Rusted    Pipe  of   Iruii   or  Light 

Mud  l>cpu.sits. 


I.  lliL-r.   l\.i;i|.cl  St  .   lu  \c.ir-;  uid    

^.  Hull.   Si  hiiiu  i'^.  I  ■(  \<'ars  1  dd 

V  Cniliilrt    

4  I  ),ni  \'.      Series  \2 

5.  liieii.  S(iiidutL:.   I  •;  ve.irs  old 

(1.  l'".ii)tiiii^.  I  iisteil  pipe  .... 

7.  D.iriy.     .Sei  ies  14 

S.  Ihcit  ■'       1 3,1.  22  years    . . 

S.  I  lien.  Stiiiiih.ius,  zz  years 

i;.  11)111,  C'.ir.iliiKii.  15  years m  — 

10.  D.iK  V.      Series  1 1)        

I  1,  t'oiiplet.  

1;.  Ihen,    !<■  H|h:i  b.iiiin  , 

IJ.  I'!llill;inil 

\2  Ilieii.   I  lei(triik.im|iswo|r,  25  years til 

p  ;.  INell.   1  l.linm  St 

14.  Il)ci!.  Cilacib  Cliau.ssee.  ... 

14.  I  )tincar; 

1 1;.  H.iilf\- 


»«    :=: 

.OH  37  5 

m  — 

•1-47 

m  — 

.oK8,S 

m  = 

.02945 

m  = 

.1247 

III  — 

.020.S35 

HI    = 

.of)  5  2 

/«  = 

.2502  + 

>/t  = 

.2502 

i/i  — 

.2502 

III  — 

"995 

III  = 

.266 

III  = 

•25 

///  = 

.2075  + 

til  — 

.4167 

ui  — 

-'S 

III  — 

250 

III  = 

■  :>  + 

III     IT 

41 '.7 

i;o 


i-irr-i i.oii    III  K.K  iu\. 


I 


H,.     I.,..l,,-     

i~.  Siini"--''!!.      Srrics  3 .... 

iK,  l,r-iir , 

Ii;.  ("•.uplft 

20.  Simpson >" 

21.  (irct'iic 

22.  McF-'lrov.  HiKiiklMi  niaiii 

2v  Slii'irfid.  l'c(iii;imu)ck  111.1111    

23.  Slit'ircni.  "  "      

',0  ■.cri"s,   !•;:  I'xpcrimriHs. 

i'ottind,- :    I'.^rlii^lit  lui iil.ii ;.  ms  7'        >;;  to  1 -,2  ";■■"/■''. 

I"  .1  li'a\\  tiilHi'iil.itioiis  :■  —  31  lo  S'j ///•'"/•*'. 


y/; 

- 

.'-"5 

;/.' 

= 

■-3 

v: 

•3-i3'> 

tl! 

= 

.4 

Ill 

- 

•3''57 

III 

= 

■r; 

11! 

■~i 

Ill 

.7  5 

111 

= 

10 

--r- 


j£- 


fit-^ 


,»9' 


*>^'^ 
,»•>' 


■TT 


^- 


'z:^ 


FlT^GEHALD'.S  EXPLRIMENTS  ON 
THE  ROSEMAHY  PIPE 


13 


>< 


Jl.4 


.-■     .1      .0      .%      3.      .-2      .4     .n      .H      i.     :i      .1     .1;      .8      1.     -■-      .1      •'■      .S 

FlC.    87.  —  I';  WL;f  r.tM'-i  K\  prniiirnf-  "ii  ilii-   K  .>.ciii.ii  \    I'ipc. 


1.  Nor  li  pun'       t'lc.incil.  iispli.i'.li-il Ill 

2.  Hotiipipi'^     rulicrcul.itfc; "; 

4  si'rif.s,  57  cxpi'i  iinonls. 
l-'oriiiuUr  :  (  umiiciI,  asph;ilt"rt,  7'  —  tcy^m-"'!'-' 
Tubcrculatcii,  :>  —  117W*/''. 


1.00 
1.00 


I'IPh-f  !  (Hi     /)/■?(,  K'.-/.Vf,s. 


1  SI 


;    .1    .fi    .-    3.    .-■    .1     .1.    >    •-'.    .■■:    .4     .0    .N    I.    .-'    .1    .ii    .s    0 

hiG.   SS.      1  !■■«>    '  liriiu>,'h  Nfw  C.isl-irun  .i:il  Cciiient-Uiieil  I'.pes. 


I. 

3 
4- 

5' 
(.. 
6. 

/  • 

«. 
8. 

9- 
lo. 
1 1. 


i;iiinami.  Nfck   •.  Si...  '"  .0827  + 

D.iMV.     Series  If. »«  =  .o^>7'75 

Ibcn,  Wctidcn  St >»  =  -"835 

Il)cti.  llallcr  Si    w=  -1245 

D.irrv.     Scries  17 '»-  .11237 

Darcv.         "       18 "<  ~      "54- 

Ehni.mn,  SuittKart "'  -0725  + 

Russell.  Si.  I...U1S >"  ^      -5 

D.ircy.     Scries  22 "'  =  .4101 

l"aiininj;.  ccinet  t -lined m  —  .41^17  -♦- 

Frietui,  Seville '«  =  -4375 

Woods.  Ni-u  toil  i(li>ut)tfiili      m  —  .5 

St'  irn-i    Rosemary  pipe m  =  l.o 

13  series,  79  experiments. 
i'ormuhi:  V  =  i;C  to  158  w '*/'■". 


riri-iiou'  o/wcA'.-z.w.s. 


A     .0      .f      3-     .■■!      .4      ,0      .H      ■-'•     .■•;      .1      .0      ..•!      1.     .-' 
Tic,.    S<j.— I"l<iu-  ihr.iiiL;!',  (11(1,  Ni-w,  and  CUaiicil  C;i<t- 


.■I 

iron 


Pipes. 


1.  Oarcv,      S(Mi(  s  I  ^ ,  .  ,     . .    ,  »/  -^  .o2<)S-, 

2.  MotiniiT.    Ti  iT(  \ ,  J"    ;o  \  c.rs  iild iii  ^-  .1107 

j.  I  l.iriv.     S(  1  its  :  j w  =  ■o^57 

4.  Mcuiiur,  Ni>l;iiii  --'ir  Sciiu',  rcw ///  ■—  -'".^j 

5.  ('(.trill,  'r.iuniiin  in.in,  :',  \  <M"^  oid ;;/  —  .(i2^ 

t).  Mfiinirr.  C  iiiiri'r.ioii,  J  v<'.irs  olil «/  =  .1(140 

7.  I  ),ir<  \ .     Sc!  ics  jcj   ///  --  .;iji'7 

.S.  i  l.irc  V.  "        -'I w  ^;  -436 

I).  1"' iiM'^.  liriiokiiiie  main.  8  yciirs  old »t  =  .iii3 

lu.  1 1  iiinlili  t.  1  serif's,  10  \  lars  (lUl in  -  .411:!  1 

II  Mriiiiicr.  HiTcy rn  4'/-l 

1  :.  Hmnlilct.  3  sci  ies.  fi.  7,  .md  i:  y^.irsnld  _  .«  .(<-^(ii 

I  ',  Mi'iiiMfi.  I  .inal  dr  ICJi-c.  I   Vtai     'Id.    .  .  w  =  7382 

14.  Unu  c.   Hl.iiH-  \'.iiic\ .  new rii  =  1.0 

Hi   -ITU'S,   ^<1   (XIIIT  IllU'lltS, 

Ji»  niiti'.i :   ;■    -  9(1  to  14S  m'^'i-". 


I.IHI.iS    Ol-    ril'l     (.'ViHUIMS.  153 

10.  Values  of  c,  x,  and  y  in  the  Formula  v       cm'i"  — 

'lutt'iii    tciiiiii    I  .cc    l\c_\n<il(l>      tcirimila)    tli.it,    in    the    ^nicral 
foriiiiil.i  V  —  ( in'i^', 

.r -|-_;' —  a  I  nn--t,int   —   1.17, 

and  tin  itturr 

:■  =  (III  I  . 

Tlic  \.iliit-;  ..f     ,mil   )'  be  ]\\\<.  t.ihul.itrd  as  fnllows: 

y 

I'mI   till  ]il|lC 1 89       .58 

I'lir    If.nl  jii]..' i()S       .58      < 'liici    <.x[PiTiiiu  T.r^  i^ivo   I.-  = 

1 8y. 
I-'. )i  brass,  /ini,  ,111(1  gl.iss  pipe      1(15       .■-,('<     In   one   set   of  ),'Iass   ixpcri- 

inoiits  ,   —  141. 
For  uroiit^ht  it.iii  pipe 160      .5;     <■  varies   liom  i  2;  to  id;,  ap- 

proximatnii;  to  the  lusher 

iiuiiibtr. 

I'or  \voo(i-stave  pipe 1:5       .51 

I'or     ne.v    cast-iron    or    tarred 

I'ipe 130      ,;i      in  tarrcil  pipes  ;   v.ir  les  ironi 

1  1 ;  to   I  5J,   the   values  Iju 

int;  .ihoiit    tlie   same  as   in 

cast-iion     pipes     of     same 

size.      IU'n/iiii;er   i,'i\esfiir 

a  60-iri.     .  asi-iii.n    pipe    i 

=   I2.>. 
F'.r  pijie  in  servire 104      .51      ( ienerallv  ,   is  ,ilHiiit   105.     In 

the    Rosfniar\'    pipe    i    =^ 

I :;. 

I-' or  lubrrrul.ited  pipe 30  to  Ho  ,51 

I-'  r  lip  rivetiil  pipe 113       .51      .•  \  .iries   fr' .111    I2f;tii    l3;U)r 

new  to   I  10  to  114  lur  pipe 

in  service. 
For  rubber  .md  leather  liose.  .      Ko      .51 
I'"or  \vroiij;lit-iron  pipe  asphalt- 
coated  170       ,  ,;      In  some  cases,   .::;  l4o,aiid  n 

the  4S-in.  |)ipe  1    -  ic/j. 

I'"or  h\ri:;e  biick  ((.iiduits 1:9       .5:      riiobstriicted  by  shafts. 

F'or  !ari;e  bi  1.  k  conduits 91       ,;:      i"nllert,.n  Avenue  conduit  of 

Chicatjo  rt-ater-supply. 
F'.i  larj;e  brii  k  i-onilnits I  lo      ,52     Chica^;o  land  tunnel. 


i 


\  r'i 


154 


F.X.4M"iFS. 


1  lie  valiR^  i>r  (  ..I  tnis  tahK'  ;;rc  iiu'ivi  values  and  ncifS- 
>,'.ril\-  v.ir\-  \Mth  ai;.'  ami  rmii^liiiL-S'^. 

'llirnii-li'iut  the  a!ial>sH  .,!  tlu'se  cxp^jrimcnts  the  total 
head  was  iliiiiini-lu  il  hy  liie  l"-s  ,,1'  h.ad  at  eiitr.inci-,  and  i!i 
the    cr.srs   III   \\\\\yh    till-    I'.ss    Iiad  ll'-t    Itet  !l   tMUlld   1)\-  incails    of 

I      T'-' 

piczdiiuter-    It    lia-    Ikih    eakulatcd    Iimhi        -^  ^,   r,    hciiiL;   the 

'  '     -s 

t'lrUkient  lit'  I  MiiiraetiMn. 

.\--iiiniiiy  tiir  1    the  appi'ixiiuali-  value  .3,    lutlon's  toriiiula 

lieeiiiiie- 

.  -  -  .  •  .1 

•  111*1- . 


V  =cm''i' 


Y.\.  I.  Till'  heart  ovit  tlic  s1i;ii)i  ••(Ire-  ciitraiu  f  imIm  ,1  pmm'.  ii>X)ft. 
lonn  and  j.assiii):  1  eii.  ft.  ot  UMtir  |.iT  sec.  is  <)  (t  I'liid  the  (iiaiii.ier, 
lakinn  /'  -    .005;. 

-•■'i\         ,    ,    4  >    "055  >   '""«\  4')         (      ,   ,    "\ 

For  a/rt/ a|>pinximati(m,  (lisl^^;al(iin^;  thi-  first  term  on  ihi-  iit;lit- 
haiul  suit,  wliii  h  i>  small  as  rompan-il  with  the  sceoiicl  term. 


VI         --         -Vl    I 


and  ''        ;:>i'nt. 

For  a  J*-. .'»/./ .ipi"^""''"'''^'"" 


or 

and 


9  = 


40 


49 

i).   16.   121 


X   3v.S«i7. 


,/  =  .1:787  ft. 

Ex.  :.  I'hr  elf(  (  tivi-  hei^;lit  <>l  ihe  jjrartc  line  <bovc  the  crilr.ini  e  iiitu 
a  clean  ir.in  3-iti.  hram  h.  i.xx)  (l.  Iimi;.;.  is  io  It  <;  ins  How  many  |.<i>- 
I>lc  will  the  braneii  supply  with  20  jjallnns  of  w.n.  1  p.  1  ti.ad  ikt  day  o( 
J4  hour-  ' 

/        .005(1    +   ■  ,)        .,   ■ 

\  12-1/         "S'J 


rx.iMri.HS. 


;3 


and  =  J:; 

'I'iic  delivery  m  i  u    it.  per  sec. 

_22  \[\\''        _   II 

Tlic  (iclivcrv  in  ^;all'iii>.  ]"     (l.(y 

"    f  (>\    X  Co   X  ^Ki  ^   :?4   =  <)2.hi2\, 

'■4 

and  till-  iiimiljiT  ..|  ],.■., pie  ■-(•rved  imm  d.i\    -        ^     "   --  4'i40„ 

rir  4'i4'>. 

Kx.  V    I'lnd  llie  |)rci      r  il  .iiiieti  I  c.|  ri  i  .hil;!!  1"!"'  '"  ■^'^■"' '''"■"-^'""•'^°°  "^ 
uallwH"  i-Vcrv    M  lhiut~.  till-  sl'ijir  'j!  the  |>i|)e  bi'lili,'  I    in  Stx), 

22(/'  6(J.OO(5.(XXJ 

7    ,  '    "*   (  I  .  (w  .  (xj    ,:4' 

1 40CXJ 
or  ""'    - 


r>-iiiL'  ilai;eii  s  loriiiul.i.  vi/..        =  =  cr—  . 

"^  /,  <f  •  hOO 


and  t.ikin<. 


IIUI       .1    ==    1.1. 


1)007/  14U00  y 

_/  i40oo\' 


,,r  </"   -    81X)   '    .'«*>: 

Tlurcloro  </  =  i>.22  ft. 

Ex.4.    What    should     De    tlie    -..M].e   of   .1    Jt-iii    w.xnleii-st.ivc    [Hpe    to 
jjive  5.1/40.CVXJ  ^.dlotis  per  «i.iv  ' 


.'ind 

Ta''.!'  tlir  fortmil.i 
hy  the  Table. 

Thefflorc 

unr] 


.•2(: 

r                  i.v40.oix.           ^^ 
I                6^  .  24 .  60 .  60 

7'  j=  3i  (1.  per  M-c 

f  =  («/'  '"     'P. 

«■  -=  125     and    /  -  .51. 

(■  —    f«"  M  \  or  alHiiit  ."■•  'D  10.000 

!>'> 


/A.;.vvw/>vr'.v  o/    /  \i  U(.) . 


II.  Transmission  of  Energy  by  Hydraulic    P'-essure. — 

\.v\.  (J  cu  It  111  w.itii  ]i(r  '-ii.oiul  1)1  ilrivin  thr(iiiL;li  a  \<\]»  <'l 
(li.imcti  r  ,/  It  .111(1  It  iii^tli  /  (t  iiiKlir  ,i  t..t.ii  lu  ,ul  ol  //  It. 
v\Kii  Ki  //  |ni   I  (lit  III  th<    total  In. 1(1  he  .ih-iorlx  d  ill  ( i\ci  i  <  (inin^ 

tin     tlKtl.ill.ll    It  --l-t.Uli  t      PI    tllf    |(l|l(  Ihcii 

tilt     111  .1(1    1    \|)t  (idt  (1    111    li-.t  tnl    U(rrk       _    //  _    // 

=  "['  -   ' 


,111(1  tllf  I rrK'iiiu \- 


//        //  _      ;/ 


./\L:.ni), 


SiiiCf  (J  =  ,"  ,  .'iiil  J,"-  is  assuuK  (1  to  \h-    :,2, 

4 

a'i(l  till'  ict.il  .i\  .iil.ilil(    \\(iiK   111  |(  "it -|i(  iim(l-<  j.tr  scttnul 


-  -(?//-  ;;v 


nf/M" 


It'    \    i     ll'i    11 1  nil  1 11  1    ..|    li.ii  ,1  -|ii.\w  1     ill  li\  (It  (I    .It    tin    t  n.l  of 
till-  iiip.  . 


wQH  ,  n  \^       I  /  n  nH  d 

550  ^'        100        28*/       100'' \       fL    • 


nil  ((|u.iU((ti  .L;ivmi;  tht   <li->t.iini    L  to  \Uiii  li  N  li   (■;(•- power  can 
bi    ti.iif>iiiitt<-<.i  witli  .1  i<)s>  o|  n  i>i :  (tilt    ■!  lilt    tot.ii  luMil. 
Ayaiii, 


,       ..  h  i'\.  V 

tkc  cM,.  iiiK-y   =    I    _  ^  :-  ,  ^^j,      ^ 


2fLw  v- 


TR.'1\SMlSSIO\    ()l     /:.V/;A'(;>-. 


'57 


p  I  =  wH^  hciiv.r  the  iir!'--^urf  CMnr, |)-.,iiumL;  Im  ih.  IumiI  H. 
Mil-  i.ll!citncy  (limiiiislus  a-^  :■  iiu'reasrs  and  tluTcforr,  so  far 
.1^  cltk  ifiRV  is  11  pi.riTncd,  it  is  a(l\aiita;^fci)us  {<>  transmit  rtUTLjy 

It  a  Inw  spci'd.       A'Min,   tlu'  I'iticirtu  \-  is  Li>nstant  it'      ,  is  con- 

slant. 

AsMimiii;,;  tlii-  (•    he  tlic  tasc.  take  r  -  —  (-./<,/.      Thru  tlic 

total  i.'iutl;\'  transmitted  —  :<•( V/  —  :.■         : 7/ 

4 

It  It  !)(•  also  assumed  that  the    ihukni-s,  /  of  th,-  i)ipi--nirtal 
is  so  small  that  the  t^  irmiiia 

/</   -   2/7 

holds  true,  /  ■    hrin^    tiir    mi  umtcn.ntial    sticss   indiKcdin    the 
Miftal.  thi'Ti 


7T  I 

tll(     IIRT^^)     tl.lllsiniltcd  f(/l 

rr.  r  i,i 


cfV 


»  pd 


V  iiritu;  th'    \olinTic  of  thi-  pipe  per  unit  ot  Irir^th 

iinur.  toi  a  ^i\(ii  \oliiiiu  V  ol  nn'tal  ajm]  ,i  (  otistant 
'tlu  K  IK  \  .  till  (■ncr}.;\-  tiarsmitf.-d  i-^  a  masiimim  uiu-n  pd  is  a 
nil  Mimiin 


It   /■    Is    iiuicasi-d  Iiryoiid    i    i<itain    liiinl,  tin-    latio        is   no 

lonjJiT  small  and  the  lliu  km  -,s  /  wiil  lia\.-  a  'MimIii  \aliu-  than 
that  (,'ivci;  l)\  till  ciin.ition  /■  7  •/  /  I'lun  tlu  i  ost  ot  tho 
pijH-  will    also   UK  r^  u(        On  ilic  otli.-i  hand,  it  ,/  is   increased, 


1S8 


TR/INSMI.ssJOX   Ol    i;M:l«jy. 


m 


hi; 


the   ratio  .iml    thcirtV.rc    alsn   tlu    iir<"^'<nri'  f.    will    n-:    :iin 

u 

small,  anil  thn<  thi-  lost  i>\  thi'  ]>iiir  will  HMt  itu  rcasc  llcnce 
it  is  men.'  CLciiidniical  to  Lni])l(>y  l.iii;c  pijics  and  low  j)rc.-.siires 
than  'Miiall  pipc^  and  hi'^'h  pressures. 

I  111  ilcniand  tor  In  lir.iiilii.  pourr  in  larL;c  iiti(  ^  lias  K-il  to 
the  la\in;_;  (l<i\\n  ot  nL'tworks  ,>\  niam^  tllinu.h  wliicli  'vatri  ' 
L'on\i._\i(i  under  pressure  and  is  d.strihuti-d  to  the  con-iiinci 
for  various  industrial  pur]iosos.  Siiui-  tlu-  loss  of  ';ca>.  due 
to  iVieti'iiial  ri-~!-,t.nu  <■  i^  appi-o\iniati'ly  ])ropor' onal  t'l  the 
S([uare  of  tlu'  \i  loniy,  .md  ■-nue  .d-o  ihr  iiionu  uni  of  i! 
ii)()\int^  fluid  must  not  he  so  L;reat  .is  to  make  c.\eessi\-e  -^ii  'cks 
possjlilc,  hi;di  \rlo(.ities  cannot  he  allowed  in  the  mains  or  in 
till-  niaelune^  operati-d  hy  thi-  pressure-water  except  for  vcr\' 
sli(irl  di--taiii  1  .,  riiu-,  the  \('I(Hit_»  ol  llow  in  the  mains  i-i 
IiiiMti-  1  to  I'  )!.  pel  sriiiiid,  and  rarel)'  exceeds  S  ft.  per 
st-eond  in  tin  inn  hme>.  In  London  the  avcraije  rate  is  4  ft. 
per  second.  .\;^.nn.  the  ijKiuililv  of  pnwcr  con\X'yed  b}-  a 
single  maini.mnot  In  ;..;re.it  Ihiue  ilu  hydraulic  distribu- 
tion of  i>o\\i-r.  in  u  hii  li  the  ])rtsHiire  ot  \^  .iter  is  direct  1}"  utilized, 
is  especi.dly  .ulaptrd  tor  machines  with  -.lo\\ -nioviii^  r.uns, 
w  hii  h  are  inttnnitient  in  .u  ti"n  and  \v  liu  ii  \\  ork  onl\  tor  '-h'  ii  t 
intc  r\.iU  ot  time,  as,  (or  exmple,  in  littmi;  ,ind  pressing;  opera- 
tion, .nid  wiie'i  a  14re.il  ellort  is  in  be  e.xertcd  throuj;!!  a 
'^IkpiI  di,!.nue  In  London  the  pressure  in  the  mains  is  7:^0 
111--  |ier  s,[  in  ,  but  in  the  nmre  recent  distributions  in  .M.m- 
chester  and  (ilas^ow  the  pressure  is  1  1 00  lbs.  per  stp  in.  I  he 
working  stress  in  the  c.ist-iron  mains,  the  largest  in  use  bein^ 
•j\  ins.  in  diameter,  is  2S00  lbs.  per  si],  in.,  and  they  ,ire 
;.;en(r,d]y  te-tcd  to  2300  lbs  per  s(|  m,  lictore  l.uii:;;  ,ind  to 
about  icxxj  lbs.  per  stp  in.  .ilter  l.iyini^  1  he  thickness  /  in 
inches  of  c.'ist-iron  m.iin  ot  ,,''  ins  dianutir  under  .1  w.iter- 
prcssiirc  of/  ll)s.  per  s(|.  m.  m.i>  In  deteniiuietl  by  liie 
forniul.i 

/  =  .0007 S/i/  -j-  .::5  in. 


i:x-iMr/.Fs. 


'59 


AnotluT  form  i  ^'ivcs  /  ■ --.  ooz^f'i/  -}-  .73  in.,  />  !)ciii;^  tlu' 
pressure  in  alnii)s])licrcs. 

\\  ith  suitable  joints,  and  drawn  tuht's  cit  stcol  with  a  tcnacitN' 
<)t  15,000  lbs.  ])or  s(|,  in,,  tlif  lixiliaulic  s_\  lUin  ol  distribution 
could  !)'■  urcatl)   i-vtciuK'' 

Aj(ain,  tor  .111  hydr.iuliL  \i\\w  or  press 

.    /..'-o*  -r^>:  ,  A.  -  ry    '-.r'y 

wlicrc />„ .  /"i   .!!(    till-    inUii-itic--   >>{    pressure    .it    llu-   outer    ,i,u! 
iniuT  surl.Kcs, 
/  is  tlu-  iiitti'-ity  of  stress  at  the  r.idius  r; 
>'„ .  '■,  -ne  till    i,ulii  ot  the  outer  .iiid  inner  -lurfices. 
(Sec  Appciuli.\,  Hovcy's  ••  Theor)- of  Structures.") 


V.\  I.  All  .iccMiiUilalor  supplii-s  a  pressure  tit  700  llis.  per  sq.  in. 
What  length  iif  H-iu.  pipe  will  dtliver  :  xj  11.1'.  of  iisi-fal  ciierjiy  with 
1  loss  1)1  :o  prr  1 1  III  ^ 

J5>>  U.P.  enti  I  till-  pipe  'riicicfmr,  if  O  is  the  (ieliverv  in  i:a.  ft.  of 
Wiuer  pt:r  set., 

i-jl    -00 
2  50    -  <J, 

55'J 


252 


=  Q 


"  :(;)■- 


and 


V  =        It.  per  sec. 

o 


r.iki        /        001;  [1    +  -    1  =-         ,  fcr  a  elpaii  irem  pipe. 

^  ^  \  r  J   X  1/        |0<X)  ' 


Hun 

'°     .   .50        I..SS       6.i  .  -'"5   .  4   X   tA.JI^  (V)'  ... 
UH'  .'5J  I  64      550 

J.  Ix'iMi.;  the  l<  iil;i1i  i.f  tlir  pipe. 

111!  ti  toi^  /.  =  78,293.7  ft.  —   I  t  S  niiloi. 

Kx    .:    Tlic  ("(ricietav  of  an  <mi);iiu'  is  />,    it  hnrns  z  Ihs.  of  roal  per 
hour  per  H.I'    ami  works  id  li(»urs  a  day  (or  300  <lays  in  the  year.     The 


IT; 


\K 


\u 


lis- 


n. 


f*i' 


1 60 


rKi:sst  h'f-  iH  I    /«'  >h<)(:k. 


cost  ..I  til-  cnj;uK'  i-.$i:pci  1 1  I' ,  .nui  tin-  ro^l  .if  1  lip  <  d.il  S^  per  toti. 
An  amount  of  4f;oo  j;alloiis  ot  sv.iti-i  pn  ininiitc  1^  lo  In'  i.umiI  1  viHk  .iI 
lieiL;lit  ol  :00  ft.  What  imist  !»•  ilu-  mmiiiuuii  ili.iin,,  /'.of  tlir  \'\\><\ 
as<iiniiii^  tli.it  thcio-t  of  till-  pMiiiiL'  IS  S/'  i»i  iiiiial  loot.  ai:ii  ili.ii 
/■  =  .00(14  ? 

Ia'I  //  tret  hi'  tilt:  fiictional  loss  ot  li'Mil. 


TiK-n,  since 


4500  22    /'" 

60  X  ()J  ~  ;     4 


■/)■ 


I  OK 

I I 


,4    r 


4     X      ,0*3^14     .-      2(.K>    I     1  '.Y' 


/> 


.4       /'' 


2  /Kisy  I 


\i;;\m.  k-t    .\"  lie  the  mmibrr  of  i  I  [', 


Tlu-n 


5    '- 


"-(J(XJ  +  /■) 


^5  * 
=  ,  ?uo  + 


25 1,77  j  /''  » 


2  .  I'l .  30,).  ',      lOi. 
Cost  (if  (  oal  capitalized  at  5''     ;  .V.  _^  ■ 

Cost  of  onfjini'  =  S'*\- 
fist  of  piping  =;  $200/ >. 

T  'i.il  01  line  cost  —  ,30oA'  +  200/' 


!;28SA'. 


* 


=  300  X  15  J  -""  + 

wliii  li  ninst  be  a  ininininm. 


»  .;oo/'. 


+  200, 


an'! 


/•  ^-  3..>X  ft. 


I  ii-inc.  .ils' ..  // 

No.  of  II. 1'         \'      454-y' 
(apit.il  cost  —  $I37."73- 


11    '  i_voXr 


^  .01868  ft. 


12.  Pressure  Due  to  Shock.  Wa'.'v  iImu  -  tlnonoli  .1  Imc 
111'  iiiiiitio  \\i!h  .1  \(liH,i\  III  .  tl.  |>f'  'I'luiiil,  .nil!  ,it  accilaiii 
jxiiiit  till-  miitiiiii  is  siuldciil)  .iiiistcil  by  t!it  ilnsmo  .it  .1  \mI\c. 
<k\ilii|imo    ,1   siuldcn   iiiircasr   in   tiic   pressure  .U   tin   \.il\t    ot 


rf<nSSLKI-    Pit     ID   SHOCK.  "" 

/"lbs.  per  SI  I .  in.  W.itcr  liein-  --liL^htlN-  <- omi)rc-.sil)lf  ^l'>-~iii^' 
J„  u\  its  l)ulk  uuiliT  .1  prcsMiro  of  ::  tons  (of  ::240  lbs,  i  prr 
siiuan-  iiuli  -.1  i.Mriq)rts.,i,,n-u-,ivr  st.irts  frdin  ih:-  \\i\\v  .mi 
nuuTS  backwatiU  tin.  ui-li.uit  tlic  wln-l''  !rii-th  A  ft.  'A  tlu; 
nioviiiL;  cnlimiii  of  u, iter.  'IIk  w.itir  still  (iinvs  th<  pipe  \'<r 
the  period  lit"  /  -ei  mills,  (iuriiiL;  uhieli  the  compression  con- 
tinues. 

Let  ,1  It.    I>e  the  sccti'Mial  area  "f  tlie  water-cohiniii  ; 

"      .1   ft.  be  the  (liiiiinutiMii    in    tin     leiv^th    /    of  thr    \v..ter 

column  ; 
•■       A' b,  th<   in.uhiliis  .il  ., '.ihic  I  l.istuit>  ..f  water  =  300,000 

lbs.    per  sq.    in. 

Then 

X  _  / 
/.  ""  JC 

X  =  7'/, 


1,1,1        14. p?  ft  munKiniini  nt'ihe  fluid  mass  —     ^aL~'. 


I  lelKC 


144.^/'    ~  U4.r  /''*  ~    '44.'.'^   /'  ^"' 


and 


=  vcloeit\-  nf  the  \\M\-e-i)r<>p;i!.;at;' m  -■  t 


v/=:'- 


Substituting;  the  values  of  _<,-■,  A',  am!  r.',  the  vel"cii>  .4  va\e- 
propajjation  is  fouml  to  lie  about  4"20  ft  jier  ^'cmil,  whkIi  is 
also  the  velocity  i>f  sdimd  in  water. 

Ex.  A  volume  of  water  5.1  (l.  in  length,  flowiiij;  ihrmicli  a  pipe  with 
a  velocity  of  24  ft.  per  see.,  is  (luiekiy  and  uniformly  stopped  in  ene  tenth 
of  a  second  by  ■  ios'iif-  .1  Mop-v;ilvc.  Find  the  increase  of  pressure  per 
sq.  in.  in  the  pipe  near  the  valve. 

The  prcs.  per  s(|.  in.  =  ^  ^'  .  .  -  162.76  lbs. 

3-        ' 44  ' 


l62 


i/A>u    i\  riri:  <:o\\!j:n.\t;  iiro  ki si  kioirs. 


13.  Flow  in  a  Pipe  of  Uniiorm  Section  and  of  Length  /., 
connecting  two  Reservoirs  at  Different  Levels,  i.it  ti.  he 
the   (lirfcreiiLC   u^  level    between   the    \\'ater--.uir,ice   ill    tile   two 


reservoirs. 


I'll..    ((J. 

The  work  done  per  seeoiul  is  cvidiiitl)-  eciu.il  to  tlie  work 
tlonc  by  the  t'.ill  nf  ri'^_' lbs.  of  water  thri>ii_-;h  tlie  vertical  dis- 
tance c,  and  is  e\])ended  — 

(I  )    In  pniduein;.;  tiie  \clocit_\-  of  ll"W  :■  ft.  per  second,  wliiih 

reijuircs  a  liead  of  r,  ft.  and  an  expenditure  <>f  ii'(  '..-, 

ft. -lbs.  of  work  per  second; 
(2)   In  overconiin;.;   the   resistance  at  the   entr.uue  from  tli<' 

upper  ri-i  :vMir  into  the  pipe,  which  re(]uires  a  licad 

of  r.,  ft.  and  an  expni  litnre  of  :i'(J-,  ft. -lbs.  of  work 

]ier  second; 
(.^l   In  overcoming,'  thi-   fiivtimial  resistance,  whiili  reciiiires 

a  lie, id  of   "    ft.  and  an    expenditure   of  ri'(_'.;,  It.dbs. 

of  uork  per  second.       I  luis 

or 

Now    -,  =  ft.,    and   the  corres])ondint^   ''nert;y  ;i'C'-i   i« 

iiltini.itely  waste<l  in  producin;^  eitdy  motions,  etc.,  in  tin-  lowe. 
resirvoir. 

J,  mAV  be   i'\pressed  in  the  form   11^     ft.,   v  bein^  a  eoeffi 


CH/'/yS    /().Vo-/7/7f   lORMn.-l. 


"'? 


cient  whose  value  v.iric-^  uitli  the  iialnre  'A  tin  i. .  piistiiK  tinii  df 
tin-  entrance  into  tlie  \n\K- .  If  the  |ii|)e-t'ntr,nu  i'  i--  hell-nicmth 
in  !"(>rr.  ,  .'.'  =  .OI  it  a?.,  hnt  if  it  i>  eyh'tulricai.  u  —  .49. 
l-inally. 

c,  =  tt.    -      ,  ft., 

t.ikiiv  —  — ■  =  /      .  as  is  usual  in  praitiic.       Ileneo 


z  -  ^'^(i  +  n  n- 


4fL 
d 


-'MCe   0  =  V     and   or  !>;  a-'-Mllleil  t"  he    ;2. 

V  4 

I'lir  t;ive!i  \  allies  (jf  (_'  and  ,:  a  first    approvlmatc  value  <<f  // 
nia\'  lie  "litained  fr.  an  the  last  eiiiiation  hy  ne;.;leetini,'  the  term 

I  1   _L  /,|.       Call    this  %aliie  1/  .  and    ■-iili'-tituti    it    f.  ir    the    ,■/ 
4tW 

4  //. 
in  the  teriii       ,     within  tin   hrai.  Let-,      A  scei 'i,d  appn '\iiiiati"ii 

irja\"  lunv  be  nuuie  hy  dediKiiiL;  </h'iin  the  ln.nnil.i 

and.  the  uperatiim  may  be  a.;ai!i  leprated  il  disind. 

(jeneiall'i    '-pLa'^iii;.^,     1     •    '/  i-^   u'^iiall)   \ery   sin. ill   as  eom- 

4  '/ 
i)ared  with        ,    ,  .md  ma\'  be  di'^re'jardc_(l  \Mthi'iit  cii'^r  <  A  prae- 
'  ,r  ' 

tk  .ll    imp'  'It  IIH  e. 

!  he  il  irn'ail.i  then  beei  'ines 

4fL  v' 

d    2g 

whuh  i>  l,i>i'\\n  as  (  he/)'  '^  Iniimil.i  lor  Iodl;  pipes, 


1 64 


I.OSSIS    (>/■    HI-. -UK 


En 


m 


'1  111-    term    I  -'    //    tufd    Miily  l.c  t.ikui    iiUd   ;K(.i>unt   iii   the 

ca:-i'  'i|  <li(iri  pipt's  ami  liii;!)  xclnutiLS. 

V.\.  '\  \-\i-  (iiiicii  111  r  1-1  li  vi  1  liituciii  ilie  wiitcr-siirfares  "(  two  rescr- 
Viiirs.  ccinr.t'i  ti  il  by  a  ;4-in.  |h]ji'  (\  inilfs  in  leii^jtll,  is  iy2\{\.  'I'Ik' 
pijif,  luiviiiL,'  111  t-n  111  UM'  fur  Miriif  tinn',  li.is  it^  iiisiilf  Miifaie  <ii,iti-<l 
witli  .1  ilcpDsit.  .mil  nil  -prcuil  provi-ii.u  is  mail-.-  to  (liiniiiisli  llic  rcsisl- 
,im  r  al  the  ii|i|pei  iml.  I)clcriiiiinj  tliu  (liNcliiirj^c  iiUu  llic  lower  Ii;m;i- 
\i  lir  in  i^.illons  pi-r  Imur. 

1        I  1 

1    4- 


Take 


/■  =   .ol  [ 


Tlu;n         ir^i  =  .     ,-    +   I     *- 


12-2/         yd 


X  689, 


and  ;■      4  ft-  l"T  si'c. 

Thcrcfoic  till-  iliscliar;:!'  in  <  11.  ft.  yvr  Imur 

=  "  .  '   .  4  .(■o.(>o  =-  45,::57l, 
/       4 

;iiiil  till--  ilix  liaryc  !ii  gallons  [kt  hour 

-  45.257!   X  '■'i  -  282,R57>- 

14.  Losses  of  Head  due  to  Abrupt  Changes  of  Section, 
Elbows,  Valves,  etc.  \\  In  n  tin-  \ilin.it\  .  m-  tin-  ilncctiwn  nl 
ni'iti'ill  111  .1  "iia--  lit Watii  lliiwitiL;  thimr^li  a  ])ipi'.  i--  ahriiplly 
i.l)anL;i'i!,  \\w  water  i-<  iii'ikin  up  int"  iililu-^  ui'  irrcLMilar 
iiiiitiiins  wliiih  all-  ^imii  ilcstii  ix  1  d  I)_\  \iMii>it\\  tin  1 1  irri_-~p' nnl- 
Iiil;  fiR-rLiy  liii'i._;   wa^ti-i!, 

(■\~-l  1.  Li'ss  liia  '('  i!  ^Uiidiii  ^fiilrdilitit.  ^^.\lt,  i", 
Chap.    I.; 


I'll..  "I. 


Kii;.  112. 


{a)   T.<  t  u.iti  r  \\"\\  fmin  a   pipe  ,'^I"i^.  'jl  1,  nr  frmn  a  reser- 
voir iV\\\.  i)J'  iMti>  .1  ])ipe  i>f  sectional  ar.-a  A. 


i.ossrs  nih:  to  stioDhis'  co.vt/?.-/'  r/o.v,  rtc. 

I.ct  (■    \tc  the  Coefficient  of  contraction. 

'1  !ifii  tile  area  of  tlie  contr.icteii  -ection   -:  ,  ./,  and 


tlic  loss  of  lie 


/ 


2  AT  \C,  I 


2.'.'  "^c 


/ 


'"-'? 


where  in  =  (  - 


'I  he  \alue  of  ;;/  h.i-  not  been  fleterinineil  witli  any  L,aeat 
decree  ol  accurac}':  l)ui  if  ,  :  ;  .O4,  tin  n  /;/  .\iii.  i  lie 
value  of  c,  is  sonletiIIle^,  obtanied  from  tlie  tonnula 


\  ' 


\\  lu  11    the    ualei-    entei^    a    (.  \  liiuirica!      not     hell-niouthed) 
I'ip''  Iroiu  a    lar-<'    re-.er\oir,   the  \.ilue    of 

\ ^     II  III  i-  ahout   .;'t;. 

Il -_;  1  .<  t     till'    water    llow    acro'.;^    tlic 

ahi  iipt  JiaiiL;'   <if -(  I  tion  throuL;li  a  I  tntral 
''3'  orili.  I   111  a  diapliiaL^ni  pLu  ed  ,is  in  Vv^.  ijj. 

la  t   ,?  !)■    tlic    ,iiea  '•'i  the  orifice. 
Tlien  <   <l  i-  the  arc  .i  ot   the  tontr.uted   -^I'ction,    and 


llK 


.1   ile.n 


1  /''  V 

where  in   ---.  \        —  I  I  . 


MICROCOPY    RESOIUTION    TEST    CHART 

.-.■   i  MOCJ  IbO  ItST   CHART   No     ? 


1.0 


I.I 


25 


t^  1^    12.2 


Ml 


2.0 


1.8 


1.25    II!  1.4 


1.6 


j=     ,>1PPLIEn  tKA>1GE     Inc 


166 


/o.s.si.s-  iHJE  TO  sinnhs  costk    rnosi.  etc. 


,\i  i  onliui;  III  \Vci>l),uli. 


if 


•J 


I 


.614 


•3 


.4  -5 

.610  .607 

y.6i2  5-5^> 


if 


.8 


■9 

■59^ 
•  734 


I  .CXI 

•  S9(> 
.4« 


t;=       .616 
«/ =      231.7 

A  -^         •' 

^     =         .605  .''>03 

;«=    ',077       1.S7'''       '-'^'y 

(,-)    .\    ili,iiihr,i'.;ni    uith    .1    n-iitral  -^^ — 
orifice  of  iirt-a  ii.  pl.Ktd  m  ,1  i>liiulri-    \ 
,  ,,]     pipr     1.1"    -.rctioiial     .in. I    .1    .i-^    ill    ^.^ 

ImK-  9.}.  '■"•    '' 

The  ••  (.ontr.iLtt.cl  .ui'.i    "  "t  tin-  u:itt;r  —  C'7  and 

thcl..ss„fh-a,l         ,^,l,,   -^J    -2.1.,,-') 


T^ 


2i" 


(Kiur.illy  w/  nui-^t  he  dct.  tininnl  hy  cxi.crinu  iit.  hut  \\  cms- 
li.i,  h  L'lVi  -^  tli>-  O.ll'.uiiu;  results: 


•*, 


if 


./ 


<\  — 

.024 

/•32 

m  = 

225.9 

47 -"7 

A  "" 

.6 

•7 

i\  — 

.712 

•755 

M  = 

1 ,7(/. 

.797 

•3  .4 

.643  .659 

30.H3         7-«o« 


.^13 
.29 


•9 

.892 
.06 


■5 

.(kSi 
3-753 

1 .00 

1. 00 
00 


Lossiis  nrr  Tc  ,-iHKrrr  i-m .■>i<(;r\u-\r,  i;i.ii(m-s.  i  n:     i<>: 

("ASl-   II.    Loss  liuf  10  ,1  StiJ,/.i!  I:>ihu\^nu.nt        ^Vv^.  (j;.^ 
1 .(  t  .  /,  —  i-Nti  rii.il  .ir«;i  nf  sni.ill  pipe 


7?  I  A., 
2g\A, 


=  ■ 

V 

I"'"-- 

- 

Fii..   95- 

Tlun 


I      j7'A.^ 


rii 


—  m 


^S' 


wlierc  >ii 


\A. 


XoTF-.-Tlic  losses  of  head  iii  f".!--''  I  '■')  and  in  Casp  II  may  be 
avd'id.'d  by  substitutint;  a  graduai  .ind  r<-t;ul.ir  (  hat)j;i-  of  section  for  the 
abrupt  chani;es. 

Cam-  HI-  /''^-^  of  Ihad  ,/n.  to  l-l/'o:os.  .Fit;.  </,,  — The 
loss  of  hc.-ul  (liK-  to  the  ilisttnluiu.-  >,iu-f.l  hv  an  tlbow  is  ix- 
prcssrd  by  Wi-i-li.Kh  iii  tlu-  form 

.,'»  brill-  tlir  .  Ibow  ati^'lc. 

Wcisbatli  (lidim.l  t!ii<  formul.i  fioiii  tin  r.sulfs  of  rxpcri- 
nicnts  with  iiipcs  1.2  in.  in  diamfttr. 

'Ihc  viloeity  '\  with  wliidi  the  water  Hows  aloii^,'  th--  k  n^'th 
.;/.'  ma\-  Ik-  resolved  into  a  inmi><)nent  ;•  with  which  the  water 
flows  alon^,'  A'C  atid  a  romponeiit  i<  at  rit^ht  antjlfs  to  the 
dirt-ction    of   :.      The    Kimponetit   //   and    therefore    the  cnrrt- 

spondii.^,'  liea.l,   vi/..    ,^.  is  wasted.       I  he  component  «  evt- 
ikntly  diminishes  witii  th.'  ant;le  »/>  and  becomes  nil  when   a 


i68 


LOSSES   IV 'H    TO   lil.fK^U'S.  HfiWDS,  FTC. 


H, 


J 


gradually  and    i  niumui  uisly  iiirvcd  l)c-nd  i-^   substituted 

L-ll)'>\\'. 


>r  ilK- 


fii;.   <)(>. 
Casi    I\'.    Woisbacli   ^n\cs  tlic  follouiiie-   imi">irical  formula 
fi)r  tin-    111-,-,  ..j'  head  at  a    bend  iti  a   pipe,  (,'<  beiiii;  the  aiij;le  nt 
curvature : 

for  a  (Uiulai     pipe    "t    di.iliuter    (/.    i>  bi^illg  the     >- * '      '-- 

radius  I  it' eur\"atut"e  n|  tlie  bend,   atid  ^'c:-  97- 

fit  =  .i::4  +  ?>.io4\~y 

for  a  j)ip<'  of  rcctanj^ular  scitioti,  .v  l)eiti^  iIk'  Icn^tli  nf  a  sid. 
i)f  the  sictioii  jjaralkl  to  the  radius  of  curvature  (p)  of  the  bend. 
Accorthtr,;  to  Xavii'i, 

//    "  i.oijS    ■    .oiSfiA'i ,, 

A  2.V 
A'   liem^'    tile    r.idius   atui    /,    tlie   leiieth   of  tlie  bend    iiuMstir<-(l 
aloii},'  tlie  axis. 

As  a  result  of  recent  e.xperinients  by  (iardncr  S.  Williams 
ami  (itluTs  (I'roc.  Am.  Soc.  (.".  V..,  May,  I'joi)  it  is  claimed 
that,  down  to  a  limit  of  2\  diameters,  curves  of  short  radius 
offer  K-s-.  lesist.ince  to  flow  thait  do  curves  of  loiiyer  radius, 
which  i-  lontrar)'  to  the  ordiii.iry  hvpothesis. 


LOSSt:s  DUF.    TO  SLUICES.  yAi.yr.s,  ETC. 


169 


Casi;  \  .  /  i)l:cs^  c  (v/'i,  .Siiiitts.  etc. — Tlic  loss  of  !utii1  in 
oacli  ot  tlic  cases  represented  1)\-  the  se\er.il  t'lj^ures  nia\-  Ije 
traced  t' •  a  cniuraction  of  the  -stream  similar  to  the  contraction 
uliiili  occurs  ill  the  case  of  an  ahrujit  chaii-e  of  section.       I'lie 

loss    may   iir  exi)res-,ril    in    tin-    form    iii\     ,  and   the    folh.uin^^ 

tables  ^ive  the  re-ults  ohtanu.-d  1)>-  \\eisl)ach: 

\a)   Sliiui-  111   Pipe  <>t    K,,taii^i,Iar    Scclifi:.      ( lM^^    ij^.) 
Area  of  pipe  —  a :   area  of  sluice  ~  s. 


I       .9      .S 


.6 


Ku 


m—  .ou   .09   .39   .95    2.0.S  4.02  <S.i2   17. S  44  ;   193 


ib)   Sluhc  III   Cvliiiih lull  Pip,-.      ^Fi^^    99  ) 
5  —  ratio  (.1  heiidit  nf  npenini;  to  diamett'r  of  pipe. 

A-  =:      I       .S75     .75     .f)J5       .3       .375       .25        .[.'5 

tn  =  .00     .07     .26     .Si      2. of)    5.5J     17.00    97.8 

(<•)   C<u/c  III  ('vliiii/iidil  J'lf,   A-"\'^.   ioo\ 

J  =:    r.ltlu   lit   Closs-scctions ; 

^  =:  an;;le  tiniuiL',li  whuli  i(,tk  is  turned. 


li.;,  9y. 


' 


Fic.   loi. 

I?"  20"  25°        3t,  35° 

.77.-'.  .Ck)2  .r)i3  .535  .458 

•2<)        "5  '•5f>  .VI  5.47  9.68 

50  55"  r)o''  63^  S2'' 

■25  .19  .137  .Ol)!  :'.,; 

W=I7.3      31.2      52.6  106  joT)  4S6  X 


I'l.;. 

MO. 

If«=        t;" 

10" 

0  J  0 

•"^5 

Ml  =     .05 

.29 

If/=     40 

45" 

■f  ■-  ,v*<: 

'>i> 

%, 


I70  LOSSES   nUE    TO   CH.-IS'CIS   OF  M'CTiOX.  ETC. 

\<i)    I'ity.'ttii-ialvt   III  Cyiiiiih  Uii!  I'lpc  \V\^.    lOl  |. 

H  =  anijlc  throiiLjii  whicli  \a!\c  is  turned. 
If  ^*  =    5'        lo       i;         :!o         :^:;  '"         ,i5"       40° 

w=.J4       SJ      .'ju      1.54     251      3.'ji      ^'.22      10.8 

lfW=    4;"       ?o^        :?°       60°     65°     ;o°     90° 
m=   1S.7      v^'      S-'^-''^      '''"^      -^'''      "?'       ^- 
I'am:  \'I.     1  lif  tall  (.;■  I'rcr  -^urtacr-k'vel.  nr  loss  (ifhiad,  due 
to    sudden    cltaiij^a's   of  seetidii,   frictional    resistance,  ete.,  may- 
be ^'rai)Incail\-  represented  as  in  Im^.    102. 


Fu..    inj 


Let   a  length  of  Jiipin^;  .//.  iinineet  tu<>  reservoirs,  and  let 
//  lie  till-   dilTcrenee    .4    surtaci-level  of  tlie  uater  in   the    rescr- 


\f)irs. 


lit/    .   '     he  len;^!li  and  I, idius  of  jiortioii  .  f/^  of  jiijie. 

••    I.\,  >\ /■■^"    ••       ■■ 

••    /-:..'■:. '''    ••        " 

•'    ^.'•. ^'''     ■■        ", 

"    «,  ,  n^,  //,,  .7,  be  the  velocities  of  flow  in  .//•'.  />'£  .   (  />, 

Dii,  respectively. 


LOSSES  inn  to  chasgf.s  or  ::p<:tion,  htc.  '7' 


'1  lu   n'MTMiir  opens  abruptly  into  tin-  pipi-  at  .(. 

■I'lurc  is  an  ahniiit  clian;4c  at  /.'  from  .i  jiipo  d' raiiiu-  r,  to 
oik;  of  radius  ;•  . 

Tlu-rr  is  an  abrupt  cli.m-c  at  (.  from  a  pipe  of  radius  /  .  to 
fine  of  radius  y.^. 

At  /'  tin-  w.iti-r  llows  tlirouL^li  an  orifice  of  area  ./  in  a 
diai)lira,!_;ni.  At  /:  the  veloeit\-  of  tile  v,  at.r  as  it  eiit- r-  tlie 
lower  rcserv.ar  is  iinnicdi.Uely  dissipated  in   eddios  or  Mirticcs. 

Draw  tlie  liori/.ontal  plane  ,/);///.'/' at  a  distance  from  the 
water-surface  m  tlu'  upi)er  reservoii-  e<jual  to  the  li<ad  tlue  to 
atmospluric  ])ressure. 

Draw  vertical  lines  at  J,   /■',   (  .   /'.   /'..        lak.- 


ah  =lo.ss  of  head  at  the  entrance  A 


cJ— 


//,' 
■''^g'' 


ifu: 


due  to  friction  from  A  to  />'    —         '  /-, ; 


due  to  change  of  section  at  />'=[   '  -- i       '  • 

r'  2g  ''  ■ 


due  to  friction  from  />'  to  t     — 


e/=   "     "       "     due  to  cliaiij^c  of  section  at  L —.310^ 


^A' 


due  to  fiictioii  from  t'to  H    ^       .    '  /,, ; 

(ny  ^        \  t     1 
— -,  ~  ' )     '  J 
c,A        I  2g 


=f'',\ 


tk—    "     "       "     due  to  friction  fiom  /-' to  A    —-       *L^\ 


kl'=    "      "        "      ci)tresi)ondiiit^  to  M 


172 


I.OSSf-.S  nUF.    TO   CHASGHS   OF  SFCTIOW  FTC. 


lhrnu,L;li  /draw  a  h'lri/oUal  i)1,u!l-/.i.  Thi^  plane  must 
cvitiontly  be  at  a  ilistaiicc  tr.)m  the  uator-surface  in  the  lower 
rcscr\-oir  ('(luai  to  the  pressun-luMd  due  to  tlie  atiiif)S[)liere. 

'rilen  the  /,'////  lo'^s  of  head   :^~   //> 

=  ad  +  a/  -T'/  +  ^h  +  kl  +  qc  -f  re  +  sg  +  tk. 


+   'r^:"'/     +^'"W     +^"W     ^'■^'"'-/ 


_  "'     i    .  -    .    /'■' 


■  '', 


2i' 


:i-"+i-:--';':--"v;-+t;{i -',.•+.;•( 


■*. 


^A'-    '    (    '-.^  r.r;^  r,r,'^  r,r:\ 


'1  In    lirolati  hue  a/utfi/g/ik/  \^  tin-  lr-<lraulie  i,'radiciit. 

I'.X.  A  dean  6-in.  pnu-,  400  fi.  lun^,  e.irit.uruii};  a  (>o  l)t'Mil  wiili  a 
i;-in.  radius,  a  <(o"  hciid  with  a  ;:-iM.  radius.  uikI  a  120'  bend  witli  a 
4S.in.  ladius,  dis(liiirj,'t's  I  cu.  f(.  <if  walor  per  sec.  inty  a  clean  12-iii.  pi(.c, 
.•00  ft.  lonfj,  wliicli  aiiain  rliscliar>{i.'s  into  a  clean  4-in.  pipe,  500  It  luui;. 
cinilainin),'  four  shaip  knci'S.  v  z.,  one  of  60''.  one  of  90°.  one  of  120",  and 
one  of  150'.  rind  the  loial  hea<i  wasteil  at  llie  pipe  entrance,  . it  the 
bends,  knees,  sudileu  <liani,'es  of  serlion,  and  in  the  straight  len^'ths. 

Let  T'l,  I',,  7'i  he  the  velocities  of  How  in  llie  first,  second,  and  third 
lengths,  respectively.     TlieM 

74(2)'"  =  '=    7    4^"'''=7   4(3>' 


i 


LiX.-IMPl.ES.  173 

and 

;6                                         14  ,                                      1  :^i    , 
7',   =         ft.  tier  sec,      t,  —         ft.  per  sec.      t,  = I!,  per  sec. 

llcd'i   w.isted  al   pipe  entrance  =    -       -      -     =  •-"-'33-  't- 

0      7' 
Tlie  head  wasti'd  at  a  ticnii  =  w*-.,  -       , 

I  to    2i,' 

where  «/-,  —  .131   -f   i..S47f  '    J'. 

-/         '■         I         " 
l-(-r  =         =  --,  "I,  =  .r4544; 

.:/•'        ::"      4 

,f  (<  I 

—  i       ,  III:.  =  .13102727 ; 

-'■>      144      ■;4 
"'  _  ^'  _  ' 

2il  (/)  6 ' 

Hence 

liead  waste<l  at  60'  hcmi  —  .14544     *■   iVo   >•  ta  x  'ii>'  =  •oi9^>3:  ft., 
<ju-      ■•      --=  .130273   X  , ".I,    X  ,',  X  (?7|-  =  . 0265303  ft.. 

120°    ••    =  .131113  X  i;;;  x  „\  x  (i^y  ^  -035396(1.. 

and  I  111-  head  wasted  in  hends  =^  .081558  ft. 

Tiic  head  w.isti-d  at  .1  knee  --  iin      , 

-.C 

.        'A  ■    .  'f 

where  nit.        .'1457  snr         +  2.047  sin*  ^  . 

I'ur  a    6(j    knee    </)  =  1 20',  iiii,  =   1. 8607 

"  i^o  i/>  =     90°,  ;«<•  =     .9846 

120°     "       (/>  =T     60°,  Ml,   —     .36436 

150°     " </>  =     30*.  tin   =     .07254 

Then 

head  wasted  at    60°  knee        i.f'X)07     x   ,'<  (",',"'  =  S*^'-*^'.' ^'•• 

(/j°     ■•  =     .9846     V   ,,',('«,«>'=  2.01853  •• 

1 20°     •■  =     .36436  X   ^t  ( ',7  )•••  •=  .74697  " 

150-     ■•  .--     .07254  X   ji  (■,»,«)■'==  •'487«  '• 

and  the  heail  wasted  In  knees  =  6.72884  ft. 
Head  w.isted  at  junction 

l)etween    ^>-in.    ,ind    u-ln.  jiipes  —  i.'|(n   -  i }  1'  —  .22778  ft.. 

.;i(.,'i26\'  .     „ 

"  12-in.    and      4-in.  pipes  =  —  =  .047B3  I..,. 

64  \  1 1  / 

and  the  lic.id  wasted  al  sudden  changes  of  section         =  .87561  ft. 


»74 


•». 


NOZZUS. 

l"or  straiuht  lengths 

lake/  =  .oos    I    +    ---jj  =    -^      f"r  6-n,.  p 


ipe, 


/  I      N       .065     . 


TlK-n  luM.i  w,,-.ti-(i 


ill  isl  length 


(1        •■ 


••   3-' 


.065 
= !-= ^      ^     -.7.01929  ft.. 

•"25  . 

■^  '  f  '-    ]  =  "O.S/SSft.. 

I  <^'4V    II    / 


ami  Uu'  iriitioiial  lo>-*  uf  licacl  —  v'-'43-9  ft. 
HfiK-c  the  total  head  wasted 

=  .0S1558  +  f>.72.S.S4  +  .S7561  +  9'-457=9  ■-''  '»').i433  't. 

16.  Nozzles.— I. i-t  a  pipe  ,  / />.  "f  Icn-tli  /and  (liaiiu-trr  </. 
U'.i.l  Irciin  a  ic^.TVoir  /'  ft.  alxne  tiic  end  /.',   I'i-.    lo^ 

/■/rs/.  1a.-1  tiio  pipe  be  open  to  the  atniosplK  re  at  /.'. 
'liun 


//    -  head  to  oveieomc  re-^istanee  to  (  iiti.iiKe  at  .  /  (=  ;/ 

^         -■> 

-\-  head  to  overcouu-  re'^i'^t.incc  (hie  tolictids,  etc.  (  =  >" ^     j 

(      4//''\ 
-f-  he.ul  to  o\eii:oiiie  trictuHiat  resistance  \^=    ^^     ^  ^j 


SO/./.I.H.S. 


-\-  head   corrcspoiuliiii;   to  tlu    \  clDcit)-  ;    in    tin-   pipr   aiu! 
al  tlu   outlet  1  T^   ^'    ] 

II  4-  /;/  -\ ;-  I  -4- 


at  /.' 


K '  I , .    103. 
IriKt    tin    hriL^lit   tn  wliiiMi    the  w.itcr   is  capable  of  risin" 


<')r,  aL'am.  i*^ 


I  -4-  //  4-  III  -|-  4/' 


/ 


StYtnit/.    Li't  a  luiz/le  he  fitte<l  on  tlu-  pipr  at  />'. 

I.rt  /'  he  tin-  velocity  with  which  tlu-  uatcr  leave-  the 
iio/xle. 

Let  /'  he  the  (li:inieter  nt'tlu-  lui/zle-'uitlet. 

'i'lii'^  diameter  is  ver>-  -mall  as  compared  with  the  diameter 
1/  of  the  pi])-.        Ihit 


■■\i\i\  therefore 


4         ~     4 


so  that   /'  is  \-er>'  larj^c  as  comp;ired  witli  T'. 


176 


NO/ZLES 


Also, 
//  -=  liead  to  overcome  tlic  resistance  to  entrance  at  .  / 

+  heail  to  overcome  tlie  resistance  due  to  bends,  etc. 

-j-  he:id  to  overcome  the  frictioiial  resistance  in  jiipc 

-f  head    to   overcome    the    InctMna!    resistance    111    nozzle 

-f  head   corresi)on(Hn<;   t-  the   velocity   /    wuh  wnich   the 
water  leaves  the  tiozzle    1  -    ^  J 


4./V\ 


r- 


m 


and  the   hci-ht  to  which   the  water  is  now  capable   >,{  risini;  at 

/y  is 


r- 
2g 


-.2  / 

2ff\ 


-  h  —  [  \n-r  t'l   :-   -,-  '-  w 


4//-. 


1 


111 


J- 


I^tf'"  .    -   //,, ,  be  the  pres'-urc-head  at  the  entrance  t'i  the 
lozzle.      Then  the  elTective  head  at  the  s;inie  point 


_-  /,   4-  -L  —  1 1  -1-  I)!  ) 

—       "     '24,'-  2ji 


Hen 


-•ncc 


V^ 


I  -\-  III    — 


/>' 


It  will  be  observed  that  the  delivery  t'ii>m  the  nozzle  is  less 
than  that  from  the  pipe  before  the  nozzle  was  attached,  but 
that  the  velocity-head  at  the  nozzle-outlet  is  enormously 
increased.  The  actual  hei^lit  to  which  the  water  rises  on 
leaving;  a  nozzle  is  less  than  the  calculated  hci^dit.  owing  to 


NOZZLES. 


177 


air-rcsi-^tance   and  to  tlie   inipatt    ni   jiartirlc<;  of  water  a-;   they 

tall  b.ick. 

The  I'orcc  ro(iuiri-il  i.i  hnld  the  iv././ie  is  evidently 

"  ^  J'    _  J/2 

,?-  A'     4         ■ 

If  the  wat.  r  tlmviiiL;  tlirou^'h  a  pipe,  or  hose,  of  length  /ft,, 
with  a  velocity  of  :■  ft.  per  second,  is  quickly  and  uniformly 
shut  offh,-  a  stop-valve  in  /  sec,  the  pressure  in  the  pipe  near 

the  vaK.'  is  increased  hy  an  .iniount  lbs.  per  S(iuare  toot. 

n\  two  iMims  (if  nozzle  in  j^eneral  use,  the  one  CFig-  IO5) 
is  a  surface  of  revolution  with  a  section  which  },'radually 
diminishes  to  the  outlet,  while  the  other  (Fig.  104)  is  a  frustum 


F!'..  I'M-  ^"'■-    '"5- 

of  a  tone,  liavin-  a  diaphragm  witli  a  small  circular  orifice  at 
llu-  outlet.  Denoting  the  former  by  A  and  the  latter  by  />', 
the  toll. .wing  table  gives  tile  rc-ults  of  Idlis's  e.xi'.erinient'^ ; 

Hrighl  of  jet  from      M(t(;ht  of  jet  from  .Hcit^ht  o(  jci  from 

i-mrh  Nt'Zile.  li  inch   Nnziie.  ij-inch  Nozzle. 


•rr^siirc  in  lbs. 

He.Tl  in 

l.er  s,j.  in. 

(cel. 

A 

10 

2\ 

22 

30 

40 

4.1 

30 

6.) 

62 

40 

()2 

7q 

50 

115 

'14 

60 

'.I' 

I03 

70 

if)i 

121 

80 

1S4 

i3t 

go 

20: 

140 

100 

230 

14s 

^ 

A 

n 

A 

22 

22 

22 

23 

42 

43 

43 

43 

f.I 

63 

62 

(>3 

;8 

St 

70 

82 

92 

07 

94 

9" 

104 

!  12 

loS 

'  '5 

>     115 

125 

121 

I2<) 

1  124 

'37 

I3> 

142 

!   n2 

148 

141 

IM 

1   isf" 

>5> 

149 

J''4 

22 
41 

63 

80 

95 
lU) 

123 

135 

146 

155 


The  coefTfKients  of  discharge  for  smooth  cone  nozzles  are, 
very  .i])pro.\imately,  .983  for  .1  ;;-in,.  .<).'^^  toi  a  ;-m,,  ,072 
for  a  i-in.,  .976  for  a  1  J-in.,  and  ,971  for  a  i^-in,   no/zle. 


178  T.lliU;  c)/    l-'RtCTU^S.-ll    /ilSSFS   !\  f/d^F. 

Frocin.m    pi'ipn^cil   the    i^-in.    iU)/;/lc   slii.wn    b\-   I-'i;^.    106 


Fk;    106. 

as  a  standard  with  a    coefficient    of  discharge  =  .97,"       Tlie 
coefficient  of  discliarj^e  for  a  s(iuare  rin^'  no//lc  i-;  about  .74. 


IKKIMANS       rAlil.l".     SIIOWINC.     COM  I'AKA  1 1\  1.     IRKriDNAI. 
I.DSS    IN    VARIOl"      KINDS    or    IIOSK. 

The  <-iini|);iris()n  is  made  on  the  basis  o(  a  f\ov  of  240  gals,  prr  iniii.. 
which  is  ab,>ut  the  ()uantity  liisrhatucil  by  a  l4iii.  imz/lc  uiul'T  a  prtssiire 
of  4"  lli^.  i>ir  'iqiiar'"  itn  h  at  ha^r  ■>{  nlavpipe. 


Ch.ir.ictir  cif  llosr 


2       —  1 


9  u 


■o       ic     '   1! 

I  lit  H^rt^iisl''  1^ 


*». 


ai"  sului  riibhrr  hosr.  cxira  heavy,  sinu'th 

ami  (trr  f mm  ruic*"'*        ■ 

ttl"  S'li'd  iiihhirr  liosr  Imhlrr  Ih.m  pmcdtnff' 

.lint  not  so  orefii.  y  m»df* ) 

at*  \\<.v<n  i-dttMi  h-nif.  ritlihtT'Uned.  reffular  1 

hi-rtvv  Hrf-di-p.iTlim'nt  h-mc 

^t"  wivt'n  foit'.n  ii'se.  r  ihbcrlincd.  Itjihirr 

thtn  pretfiltnu.  hul    of    iibuut    thr    same 

*-mti"(hnrM  of  inirimr  . I 

•4 "kniiiotton  hoHT.ruhhrr-tined.  A  medium- 

Wfiylit  host- - .  I 

Iff  knit  cotlr>n  ho%r,  riibbcr*Iincd.     Intrrinr  | 

mr^luim  smooth  .        .  I 

9^'  kiiti  loltiin  h"<ic.  ruhlkff  lined.   A  regular 

firi'-^trp^rtment  hoHT .  

•4"  knit  cotton  hn4r»  rubher-lincd.     Intide 

father  rouirli 

b("  knit  (oitnn  hone.  ni>bertinrd.      About 

^Aiiie  is  pret  edintf,  hui  a  hll!e  bedvirf 

74"  If-tthcr  !n»*r . 

tj"  W'tven   t"Hon.    nihtH-r-lmi'd,    miU    hoM. 

Mf«l  urn  fhtn  rubbrr  hhinc  .   . 

»i'    unlinrti  linrn  h  'vr  ... 

t''  W"v*n  rntton.  nibhrr-I.ned  hn« 
»|"  hnen  hose   with  t  ' «  oup   n^^. . 


*4* 


10  o 
11.5 


}     '  l..t 


4-t»    !  14  0  I  14- y* 

!         I 

-     ft  14     t         l'<       - 


• .. 

■  /  " 

t.  ^1. 

■¥*t 

•«  J 

14  i« 

tn 

•  9.4 
»•  * 

1)  « 

1*  ji 

4-« 

Ml 

nr-ni-.R-M()roR. 


179 


1  InitL  Ft  an  rni^'inc.  working  against  a  pressure  of/"  Ihs. 
I>lt  -qiiarc  t'lDt,  pump-  0  k\\.  ft  Mf^atcr  per  mcoikI  throujjh 
a  ni)/./.!c  at  llic  cm!  of  a  li.i-c  /ft.  in  Icngtli,  tluti 

tlu-  i)niiipiiv/  11.1'.  of  the  rnt:ino  =  — '-. 

550 

'I  lu;  total  lua.l  at  tiic  cnj^iiu-  rmi  ^A  \\\v  Im^c  the  head 
corrcs])()nilirii:  to  tin-  iircssiirc />  in  tlu;  husc  -•-  the  head  required 
to  proiluee  the  \elocity  of  tlow  ,- 


4- 


2.C 


and  this  head  is  expended  in  overcomiiiL;  the  frietional  resist- 
ance of  the  hose  (all  other  resistances  are  disre^'anled ;  and  in 
producing  the  velocity  of  How   / '  at  the  outlet.      Hence 


.unl  therefore 


ii'  t/     2g     '     2x  2i^--' 

^  8C-  /  t  I     ,    4./"/\ 


since 


(?  = 


,T,/ 


/'■■ 


/■ 


Thi-  pumping,'  i  i.  1'. 

17.   Motor  Dri/en  by  Water  from  a  Pipe.— Let  the  nozzle 

in   the   preceding;   artu  le   l)e  replaced  In-  a   cylinder   having;   its 
piston  driven  hy  the  water  from  the  pijK-. 

I.et  /I       the  velocity  of  tiie  piston  per  second. 


V. 


I  So 


EX.-^MriF. 


Let/',,,   --  unit   pressure   at  the  end  of  tlie   pipe,  i.e.,  in  the 
c\'liniler. 

Let  (/„,  --  diameter  of  i  yhnder. 
Then 


■/.. 


\" 


velocity  ot  How  in  ])ipc  —  y      j  u 


Henc( 


other  losses  of  head  being  ilisrcgarded. 

Kx.  A  3l-in.  clean  pipe,  325  (t.  lung,  le.ids  from  a  reservoir  with  a 
water  surface  300  fl.  above  flatuni  to  a  point  A,  1S7J  ft.  above  .latum. 
Find  (<i)the  height  to  wliicli  liie  water  is  capable  ..f  rising  at  W  d'  ii 
tlie  pipe  is  open  to  the  atmospliere  ;  (2)  if  't  teruiinaies  111  a  l-in.  nozzle. 
What  kI>)  force  is  i.quired  to  hold  the  nozzle.'  If  the  pipe  is  used  to 
supply  i)ressure  to  a  vv.iler-engine  with  a  28-in.  cylinder,  determine  (,) 
the  inaxinmni  power  whi.  h  can  be  develope.l  an.l  the  correspon.iinii 
velocity  of  tl  w  m  the  pipe.  In  the  latter  case,  what  (</)  is  the  total 
pressure  ..n  the  piston?  Take  into  account  the  resistance  at  the  pipe 
entrance  and  assi.nie  f  —  .005. 

Let  V  .111.1  r  be  velocities  of  fl.w  in  pipr  an.l  from  no^/Jc.  respect- 
ively. 

[a)    I.    300  -  I.S7J  ==  ll2i  -  tot.il  elTe.  tivc  h<  ail 


r'    /       ^  4   X   .005  X    S2>\  _"'     „i 

and    -  =  3  ft.  =  height  t.i  wIik  h  w.iter  'an  rise. 

f      rv  I  w       ^  "  .uo5j<^25\ 


f"  2985 

yi   a4oi" 


Therefore 


=  mi  X 


2401 
29«S 


=  90.49  ft.  =  hriRht  to  which  water  can  rise. 


SIPHONS. 


i8i 


{6)   Force  =  momentum 

~    32  ■  7     4\'-/  04    I  J    144  2      2985 

(c)   Let/  be  the  pres^^ui.-  in  p^.uikU  jier  sij.  ft.  .it  .1.     Then 

or  /  =  <^-J  ■  371(3  -  y. 

Hence 

3072\^         t>4A 


lbs. 


ft.    per  sec.    aiirl    tlie 


3" 

64 

max.  U.V.   =  4i85  =  4.557. 

Alsi)    /  =  6j5  .  75  =  4f'.S7i  lbs.  per  scj.  ft  .  an<l   t..t.il  pres.  uii   pi^tiMi 


;i  tons. 


Jill  tuiic.  .l/U'/K  l-'ii;.   107. 


/'I 


=  4687jx".'.Pr    '      =.0,5 

18.  Siphons.  .\  --ii)!!!)!!  i- 
ami  I-'  <i1tiii  (  inphiycd  to 
convey  watfr  Irmn  one  reser- 
voir to  aiMtlRT  .it  .1  lower 
level. 

I.ct  //, .  //_.,  respectively, 
lie  the  differences  of  level 
between  tlie  top  of  tile  sipiion 
.md  tin-  entrance  .  /  and  outlet 
/)  t(j  till  siphon.  Then,  so 
lonjj  as  the  hei^;ht  //,  docs  not 
exceed  till-  lie, id  <'f  water 
f—  32  .s  It   I   uliii-h   measures  •■"'•   '"'• 

tlie  atmospheric  pressure,  the  water  will  How  .ilotu;  the  tube  in 
th<'  direction  of  the  .irr<.w,  uitli  .1  vcl^uty  ;  i;iven  b\'  the 
e"iu.itinn 

''»  -  *'  "^    J    2g' 


-»f> 


'(■r 


182 


INriiRll/'  SIPHONS. 


I  brinL;  tlic  icnpth  of  the  tulic  .l/!CI\  and  all  resistances, 
except  t|i  It  line  ti)  trictiun.il  resistance,  beinj^  di-,rej^ardeil. 

It  //,  ^  ^J.S  ft.,  each  dftlie  brandies  A H  And  Pi  becomes 
a  u  ater-l)ar"ineti.-r,  ami  the  -iitlinii  will  no  lon^'er  work. 

I-A'en  wiieti  the  siphon  doe-,  work,  an  arrangement  must  be 
made  for  withdrawing;  the  air  whit-li  will  a!ua\-s  collect  at  the 
upper  p.u t  "I  ihi-  siphon 

19.  Inverted  Siphons. —  The  existence  of  a  cutting'  or  a 
valle)  soiiu  times  leiiiicrs  It  necessary  tMcoii\T\-  the  water  from 
a  course  .; //  to  a  Course  /'/:  by  mean-.  ■  •!  an  iiuerted  siphon 
/.V"/>  of  len,L,nli  / 

Let  //  be  the  \elocit>-  of  flow  in  .//.',  and  /i  ^  the  heij.;ht  of  />' 
above  a  datiiiii  line. 

1.1  t  ,'  be  the  \'elocit)'  of  llow  in  (he  siphon,  and  /i.^  the 
height  ol  /'  above  datum. 


Fig.  ioS 


Then 


//,   —   //^  =   loss  of  lie. id  ,it   /.' 

-{■  frictional  loss  of  luad  in  siphon 
-\-  loss  of  head  at  I) 

2jir     1^      ,/     2^"     '      2^lf 

4//  r*  .\. 

=     ,        .  approximately, 


assuiniiij^  the  eiitt.iiue  and  outlet  to  the  siphon  formed  in  such 

■■r  ','> 

a  m.imur  as  to  consitlerahlv  reduce  tin  los^e-^         and       ,  and 


AIR  /v  .-'  riPF.  ''^-^ 

t„  .11..V  of  thcs.    losses    bc.n,   d.sre.ardcd   without  practical 

t-h    a  'u-vcl,      <.cncnU!y  ^pcak.n,,    /^A    -    '--ly    ahvay.    nf 
„n,f„rm-l.l.c.      Call  tlK-lnp.  .. .       Ihcn. 
Z?/--  =  (//j  -  /^)  coscc  a. 
But 
4//  ff  _  4/  ^',Y    ,    pr ,  =,  /,_  _  /;,  =  //,  -  '>,  -  <^'a  -  ^'^^ 

,,,   ,„j,,Mon   from   wluch  /'/•  can   IK.   f^.una,  a.  ^  -  /^.,can  be 

,lct<rini'ud  bv  mean-  ^if  a  level.  ,     ,  ,  ,     ■ 

.0.    Air   in   a  Pipe.   -Th-    nVect  of  an    a,r-bubble  ,n  a  p,pe 
\i:CP  nia\-  be  lUscus-^ed  .v-  1<>11"W-; 

l.t  the' air  occupy  the  p-vtMn,  /.•(  '  of  a  pipe. 

,;,  the.urface..fthe.at.r:r.th.    re.ervn.r   supplyu,,^  the 

j„,,ebe/),it,   verncallyabove/-,an.lAJt.   above/A 


Fk;.   i"q- 


Also,  let  /.,  be  the  difference  of  level  between  C  and  IK  >> 
the  difference  of  level  between  B  and  T.  an.l  /  ,h.    tlu.kn,  ..  ■  •. 

''7"7;!:^:;^^the  head  equivalent  t.,  the  e.a.>c  rcsistan.e 
ofth.    an m  /.'(•.      Then,  .q.proxmiately. 


1 


!■   t 


184 

and 


FLOW  IS-  ripi:  OF  r.-iRviso  ^mmftek. 


/,  being  the   lenj^th   of  llu-   i)Mrtiiin   ol   pipe  from  A  to  /f,  and 
/^  tlic  li:ni;th  from  /:  t..  P. 

.\ilc!m;;  tin-  tun  (■(luatinii'^, 

/boini;  total  length  of  pipr 

Ikit  //,  —  /-}-  //^  :.-  //^  _  //  ,  very  nearly       Hence 

an  eijuation  sliowin^^   the  v.iri.ition  of   ,-   witli  a  variation   in  tlie 
hei<,'ht  //,  of  the  sjjace  (xeupied  h\   the    air 

Xiii!        //of  course  varies  with  the  trmj}(ratine. 
21.  Flow  of  Water  in  a  Pipe  of  Varying  Diameter.      Ihe 
variation  in  the  diameter  is  supposed  to  1h   >,.  j^radual  that  tlie 

fluid  filaments  tnay  still  he  assumed 
to  Il,.u  III  s(  iisililr  parailil  linos. 
(  otisid,  r  a  thin  slice  of  the 
^Q  moving;  tluid,  houndeii  b\-tlR-  trans- 
verse sections  All,  fP.  distant  .s 
and  .V  -j-  ,/s,  respectively,  hom  an 
origin  on  the  axis  of  tin    |iipc 

Let  /■  Ik   the  mean   mtiiisity  of 
pressure,  ./    the   water  area,  /'the 
wctt<-d  iienniett  r  lot  the  ^eetion  AH. 
Let    tiiese    symbols    become   />  4    ,/f.    A    \    <iA,    /'   {    ,//', 
respectively,  for  the  section  (  /). 

Let  ~  be  the  h.  i:,;lit  of  the  C.  of  (i.  of  the  section  A/l  .d)ove 
datum. 

Let  -  -t    ,/j-  be  the  hei),dit  of  the  f.  of  C.  of  the  section  T/) 
above  datum. 


Fii;.    IK 


PI 


FLOIV  IN  PIPE   OF  yARYISG   DIAMETER. 


.85 


Let  ;/,  u  -f-  (in  be  the  velocities  of  flmv  acros>   the  sirtions 


AB,  CP,  respectively, 

Then 
riif    r.ite    of 


1'' crease     o 


iiinnu'iitum  ol  til  iice 
Al'^Cn  in  the  direction 
of  the  axis 


'.  —  J 


rr  omentum      i^eturatccl      by 
the   etfective  forces  actiiii; 


UilOIl 


the  slice  in  th 


e  same 


I 


iirection. 
I  he  acceleration  m  time  at  =  ^  An  .  <>!  ,    = 


// 


.ill  .   i!lt. 


The  ti  ital  pre'-suri--  'in 


The    total    pressure    on 


A/>  —-  f'  ■  A,  anil  acts  ahmi.'  the  a 


MS 


CD  --  (/>-!-  dpvA  '\  dA, 


anil    acts 


aloiiL;  the  axis. 


Tlie  tdtal  normal  i>r 


1  pressure  on  the  surface  ACJ^P  >  >t  the  pi 


pip. 


/      ,    <" 


'>J.    ■     ''t 


t 


AC  =  2rxyf  .  AC,  Very  marly. 


The  C'lmporien 


l>  beint:  tht. 


t  of  this  pressur 


re  aion;''  tlie  axis 


=  2ni/'A( '  .  sin  n 
r=  2  T pi  .  iii\   nearl}', 

between  .  /  (    aiK.  the  axis. 


leu-  tin 


iSilil  i    lU,'//:'    [lit-  il.VlS 


Thus  the  A  .',//  I,  .udliiu!  p) 

=  pA  —  (/•  f  dp  AA  4-  ^i-^  I  -f  -  -^P>-  ■  '{>■ 
—   —  p  .  dA  -~  A  .  dp  -\-  2  npr  .  dr 
=  -A.  dp. 
since  .■/         rrr^.   and  therelorc  dA         2rrr  .  dr. 


Th 


e  com.ponetit  ol  ///,■  :etii^/tt  ot  the  slice  aii">m;  tlie  axis 

-  '.cA  .  dc 


The  fruUonal  resistnmr        P  .  AC  .  /'(ii)  -  P  .  ds  .  F(u), 
very  nearly       Hence 

^idi^-  -^  =  _  A  .  dp       u'A  .  d,  -  P.ds.  F{u), 
g 


J 


/ 


j86  EQUiy.^I.HST   IMn^h'M    V?7/V. 

;uiil  tlieretorc 


/!''- 


+  - ,  +  -v~  -^  A  ~r'^'  =  °- 


Intc;^'ratinj;, 

^  Ti-  ^  2^c  ^,  /    -■'       -■ 


a  constant. 


Take 

1  lien 


=^i 


^j,"  -t';-'* 


u- 


w 


f   Q^ 

2g        .  '    g 


a  constant. 


The  intcc^ration  can  he  (.rfti. ted  as  soon  as  tlic  relation 
between  r  .iiid  S  is  t'lxeti. 

Example. — Take  r  =  a  -\-  /'.s.  and  a'-sume  /  and  (J  tn  be 
c<in'-tant.      TIk'II 


r 


u 


+  ,;,  +  /  "'™3    /  pr    =  «i  constant, 


itid  ihereforc 


-+-       H — ,        .,     ,  =  a  constant. 

7f'       2^        4(*  gn-  ;-' 


22.  Equivalent  Uniform  Main.  -A  water-main  usually 
consists  (it  a  series  of  knL;tlis  otdillerent  diameters. 

As  a  fust  .ipproxnnatiiin  tlie  smaller  losses  of  hc.id  dm  to 
changes  of  section,  etc.,  nia\  be  disreijarded,  .uul  the  calcula- 
tions ma)'  be  further  simplified  b\-  substituting;  for  the  several 
leni;ths  .i  single  pipe  of  uniform  diameter  K'^ '"rJ  ^1"-'  same  fric- 
tion.d  loss  of  he.id.      Such  a  pipe  is  called  an  cquu.dent  m.un. 


EQUIi/AIFST   VSIFORM   .Vf.-?/.V.  '''7 

LcL  /  ,  /,,  /   Ik-  the  successive  lcnL;ths  ,,rtlK'  main. 


3 


!■'  1 . ; .   in. 


l.ct  </  ,  (/  ,  il .  'if  tl'^"  liiaiin-'teis  r 


1  ■  '  i '     .1 


be  th 


Ol.  itics  o 


if  tlicsL-  lcn;4ths 

f  ilov  in  these  leii.ml 


.tion;il    losses 


Let 

Let  h^,   //,,   //,  l>e    the   iVi 

len-ths. 
Let    /.,    ./>    :■,    /'   IJ'-'    the    enrresponilini;    quant 

e(juivalent  uniform  main. 


if    head    in   these 


itics    for    the 


Then 


and  therefore 


h    =    //,     t-/^M-/':,  + 


4/''S_4/ 


</    2 


-A  =  ^   -'/,-+ 


4.^^^,^_,4/ 


'A  + 


H< 


A',  -  Av^ + '^:;  ^ 


/, 


/, 


where  it  is  assumed  that  /  is   tin  same  for   the   several    U  iiyths 
of  tlie  main  and  also  for  the  ecjuivaient  pipe. 
Hut 

Tt(V 


7td^ 


4  4 


^  '.:,  =  etc. 
4      ' 


Hence 


J 


^'    _    A   _l-    ^i  4-  /'   +  etc.. 


an  equation  pivins  th :  diameter  ./..fan  e.iuivalent  pipe  havin^r 
the  same  total  frictional  loss  ..f  liead. 


Vfl 

! 


I 


188 


im.-INCH  M,-)l.\'   OF  UMhORM   IH.'iMhTEK. 


I'.X.  Whai  II1U-.1  be  the  (liainricr  of  ,1  iinil^.rTii  inpc  ulijtli  may  be 
siibsliiulLd  l.>ra  Inic  of  I)i|)iMK  cujisistiiii,-  ul  an  800-fl.  leii-lli  ,,{  iL-in. 
pipe  and  a  200-ft.  length  of  6-111.  pipe-? 

800  +  200  _  800       200 


rf'  = 


3''>" 


am!  therefore  d  =  .6738  ft.,  or  about  8  ins. 

23.  Branch   Main    of   Uniform   Diameter.  — In  a  branch 

ni.uii    J/.    n\    lcn_;.,th    /.    ami    dianRicr  ,/,    rtecivin-    its    -,up[)ly 
at  ./,— 

I. It  (J ,  \n:  tiu-  \\a\--srrvi(H-,  I.e.,  the  amount  of  water  L;i\t-n 
up  to  tiio  stTvicc-|>ipcs  on  ,ach  side. 

Let  n_.  be  the  cnd-srr\ice.  i.e.,  ijie  ain-.iuit  ..f"  ivater  (h's- 
eharijed  at  tile  end  />'. 

I  lien  it  ni.iy  be  assumed,  and  it  is  r>ppr..xiniately  true,   that 

the  \va_\-ser\ice  per  lineal  foot,  viz.,    '"'.  is  ci  instant. 

1  hus   the   amount   of  u.uer   consumed    m  ua\-servicc   in    a 
len;.;th  .W  of  the  main.   \\  here  /.'(  '  —  s,  is 


/. 


■(/-  —  s\ 


uhih'lh,-   t-.t.d    am.amt   ofN,,aer   fl,  ,u  in-  across   the   section  of 
the  pipe  at  t 

/.  4 

r   Inini;  the  velocity  of  flow  .it  ('. 

Nou-  ,//',  the  frictional  loss  ,,f  head  at  C  for  an   elementary 
Icn^jth  <is  of  the  pipe,  is  s„nvcn  b\-  the  equation 

M  =  -^^  •■■■'  .  ,/, 


ifiT  =  32. 


M^''  +  L  'h' 


SPECIAL   CASES   OF  riPE-FiniV. 
Intc-ratin-,  the  lot.il  i'-..  of  head  is 


)    \ 


189 


^rii  lAi.  (  Asr.s. 

Cask  I  Let  (j;  be  tlu-  total  dischar-c  for  the  same  fric- 
tional  loss  of  hcad,'/^  when  tiic  whole  of  the  uay->crv.cc  is 
stopped.      Then 


fL 


(]J 


/^  ,v^=  A  =  4^,:^^ +  (?.(?.  +    , 


Tt^d 


and  therefore 


(?:,= 


a'=  =  C/  +  (?x^,  +  3  • 


Hence 


Q. 


(         Q:\^ 


d    <  a4- 


a, 


anil    Cv 


1    .         „  /•)    _L  ^"'  qnil  ^    4-       '   .   its   mean  value 
hcs  between  (J,  -t     ^    "'"^'  ^'    1     ^  ,  ' 


heiiiLT  C^  4-  .=;s'  ';^- 

Cask    U.    ■  If  there  is  no   end-servce.  all  the  water  havintj 

1        11;.,  „  ,v  mMviee    C'   =  O,  and  therelorc  (J,    —     - 
been  absorbed  in  \\a\-sei\KL.  1^,.  ^^ 


md 


//  = 


3    ''''^'' 


Cask  111       H"  C,  -  o. 


M 


-^"'^  s-,/s  =  elementary  frictional  loss  of  head. 


tW. 


-I 


'9«>  sPijj.u.  (..t.sts  ()/  pti'i:  :t()u: 

Inte^ratinL;  bctuocii  o  and  ,v. 

3  'f"'' /.-■ 

and    tlic   vertical    slop,-,    or    lin.'   ..f  frrc    prcssiin-.    l)ccomcs    a 
cubical  i)arab'>l,i. 

^•\"l  1^  l.'t  the  main  receive  its  supply  at  J  from  a 
reservoir  .V  in  wlncii  the  surface  of  the  water  is  //,  ab<ne 
datum,  and  let  it  dischar-e  at  the  end  /;  into  a  reservoir  )'  with 
its  surface  /,■.  above  tiatum,  i'"i,L:.   i  14. 

Since  ,  (_),'■•  _  (J/  -f  a  (J:,.  ^  '-"■',  therefore 


a. 


t  +  /'-^' 


0J_ 

12 


If  0:.  ~  '  ,L\  ■  (J..~0-  and  if  (_\_>  ,  3(^;;,  then  the 
reservoir   I'will  furnish  a  portion  of  the  way-service. 

Suppose  that  .\'  -ives  the  supply  f.r  the  ilistance  .10  (:^  /) 
and  that    )'  -iipplies  /,'( '  1  =  /,,.  ' 

Let  .  be  the  hei-lit  above  datum  of  the  Mirface  in  a  pressure 
column  inserted  at  '. '. 

Then,  ne_i;'lectin-  the  loss  of  h,  ad  at  entrance, 

'..-■^=''',^'::  -{--+'■'■] 

"  ^  re/ 

~  loss  of  heat!  between  .  /  and  O  =    '  -^^IILIL 

3  ^-'f/S 


and 


..  _/a    ,   AA 


/„ 


1^.4-  ;;)-(. +  ^' 


—   loss  of  hc.id  between  /.'  and  O  =  -  ■'^u.Ji 

3    Tt^lPL^' 

Also,  /, -^-/.  =  /.. 


rR<Wl!:M   Ol'    THRr.r   RFSFRI'01R\. 


191 


24.   Three  Reservoirs  at  Different  Levels  connected  by  a 
Branched  Pipe.  — i.ct  a  \n\w  lu>  nt  lriiL;th  /,  n,   .unl  r.uim^ 

r  ft  ,  U.uliiiL;  ti"ni  a  rcser\'oir  ./  in  whi^li  the  \^at'T  staiuis 
//i  t't.  al)'i\c  ilatiim,  <li\i'lc  at  ''  into  two  !)raiiclics,  the  mic, 
('/:',  (>flcn<^tli  /,  ft.  and  ruiiu->  ;-.tt.,  It-adinL,'  Id  a  reservoir  /' 
in  which  the  water  stands  //.ft.  above  ihituni,  the  other.  ('/•'. 
of  lenc^tli  /  ft.  and  radius  /■,  ft.,  leadini;  to  a  reservoir  c'  in 
which  the  \sater  standi  '/    ft.   above  datum. 


1!" 


Fii;.  1 12. 

Let   :•,  ,    :■..,    7'^  be   tin    \elocities   of  ilow   in    /)( '.   (V:,   OF, 

respeetiv  ei\\ 
Let    (J, ,   Cj^,  C,  be  the  (.juantities  of  llow   in  f)(1,  ('/;,  ('/■', 

respectively. 
Let  ~  be  the    hei-lit   al)o\<.-   datum   to  which   the  water  uill 
ri<e  in  a  tube  in--erte(l  .it  the  junction. 

Two   ])robleni<  will    be   considered,  and   all    lossi-s  of  head 
e.vceptini^  those  due  to  hiitional  resi-.t,ince  will  be  disreL^ardeil. 

I'Kdin  IM    I         (M\en   //,  .   //., .   /l^\    /|,   )■,,   i-^;    to  fmd   ij^,    (J,. 

/■ 
(J.,;   7',.  7:,,  .■■(,  and  j.      'lakini^^  '    =  a, 


ff, 


h^  —  z 


I'or  tile  pijie  /'(',      ^~—^=a\    .  .  (i^   and    (p^  =r  Tr^'r', .  .  .  ('2) 

'1  '  I 


If); 


PROBI.I'M  OF   THREE  RESERI^OIRS. 

(3)     •■     C',  =  '^''^'8-  •  •  (4) 
OF,     '   ,   -=«;'   •  .  (5)     "      C',=  '^^,'-V  •  .   (6) 


V<>r  the  pipe    Ch, — ,      "=«  ' 


Also, 


0,-  ±c\+cv 


(7) 


From  these  seven  e(]uati(>iis  the  seven  required  (juantities 
can  be  found. 

In  equations  (3)  and  (7)  the  impcr  or  lower  signs  arc  to 
he  taken  accordini;;  as  the  flow  is  from  <'  towards  /;"  or  from  /;' 
towartls  0. 

Tiiis  may  he  easily  determined  as  follows: 

Assume  j  =  /t.,,  and  tlien  find  7\  and  t,  by  means  of  equa- 
tions (i)  and  (5).  and  hence  (.',  and  Q^  b_,  means  of  equations 
(2)  and  (6).  If  it  is  found  tliat  C\  >  (?.,  •  t'l*-""  the  flow  is  from 
O  to  £,  and  equations  (3)  and  (7)  become 


-,-  =  "'    ■■^"''    (?,  =  c\, -h  a,; 

while   if  Q^  <  Q^,   tl:c  flow  is  from  /;  to   ( ',  and  the   equations 
are 

Xiift: — It  is  assumed  that  a\—  ^)  is  the  same  for  c.ich  pipe. 

Si'FClAI.  Cask.  (Fig.  113.) — Suppose  the  pipe  O/:  closed 
a:  /•., 

Also,  let  »•,  —  /-J  —  r,  =  /-,  and  let  /'  be  tlie  m  Hmty  of  flow 
from  ,/  to  C. 

The  "  jilanc  of  charge  '"  for  the  reservoir  A  is  a  horizontal 

plane    .1A  >  distant     "    from    the    water-surface,    /"„    being  the 


rf<imi.i:M  or  ruRi't-  ri-shri^-oik.^. 


■93 


Till'   •'  jiKinf  '>t'  ch,irL;c 


tor  llic   ix-scrvdir  ( '  is  a  iKiri/mital 


])l,uie   /  S  (iist.mt      ''  Imiii  the  uatrr- surface. 


f 


In    tin-    vcrlual    liiu;     l'/\.\    take    /'.V    ;     ',,  ^  and  juin  .1/A'. 

'Mien,  ne^lectiivj;  tlic  li>-,s  of  head  at  nitraiKo.  J/A  i-  the 
"line  of  char^'e,"  or  hydraulic  ^'radient,  for  the  pipe  />/-,  and 
is  a])iirn\iniatel_\-  a  -.tr.UL;hl  Inu-. 

I  et  the  ■•  jil.me  ■ 't   ehar-e  "  A' A'  for  the  reservoir  A",  di-.tant 

from  the  water-surface,  meet  -IA\  in  d. 

II  the  jinu-tioii    ('   is   vertical!)'  below  ( ,\  there    is   no   head 

M  I  >:        -    ..h ^-• 


Fii;.  Ill- 
.ivailahlc  for   producing;   flow  cither  from  A."  towards  (>or  from 
O  towards  E,  ami  liydrostatic  «;(juilibriiim  is  cslal)lislu-d 

1(  the  ju..ctt<)n  ('  is  on  thi-  left  of  (i,  aiul  a  vertical  lint- 
-  7\//A  is  drawn  intersecting  A' A'.  MX.  anil  MQ  in  the  points 
A',  //,  an<l  /.  there  is  the  head  UK  available  ff)r  producing 
flow  from  0  towards  /:. 


i 


"'■*  rROIUFM   Of-    THRI.I:    Ki:sl:Kl-i)lKS. 

il  the  juncti-.ii  f '  i-;  on  tin-  ri-lit  of  r/.  atu!  tin-  vrrti>  ,.1  line 
^V/A7.  i-;  (Ir.iuii.  tlic  lic.id  y/A'is  n..\v  availabk-  In,-  |„..,hK,!i- 
fl'iw  tVnni  /;  towards  (). 

I.cl  the  \rrtiLal  tlirou_L,'h  (,'  meet  MO  in  /'  ami  take 
J'd    -    )         riu'ii,  apiiroxiniatel)-. 

.tiid  therefore 

II  ///.  ■     1  ,  the  ll.iu  IS  from  <'  towards  /•" 

If///.    ■   r, /:       ..        ('i 

A^'ain. 

and  therefore.  .iii|)ro.\imatel\-, 

'''    -''    --"/•^.  -/.'•         ....      .1) 

Next  assume  the   junction  <'  t,,  1,^  on  the   left  of  c.  and 
open  the  vaKe  at  /.         I  lu  ii 

/'. 

/,  -^ 

>'.           .■• 
i/^  "  ' (3> 

z/'-'':- u. 


ORiiicr  fin  ny  run  HrsrKi'uiRs 


195 


ami 
or 

■rini-^ 


ami  tluriforc 

•  .V,  +  /,)  -r  ^Z,',?-,  i-  /,:  /  -  (/,  t-  ^,)l''    -  O. 
ilciici.  assuming;  r\,  to  be  viry  -mall  as  i.  <imi)arv(l  with    /', 


/■- 


/.  +  A' 


or 


(?    -  (J         ^^^  . 


wIkTi-    ij  Tf'l  '. 


'I'liii-i  it  apiiiars  tliat  if  a  i|uantit>  (J,  of  watir  is  drawn  off 
l>\'  iiRan-  III  a  iiiaiuh  Imm  .1  111. 1111  lapaijlc  of  j,'i\iiii;  a  total 
cml-sirvicf  (J,  tliis  ciul-sirvicc  will  l)r  (liminisjicd  hy  j(_)  ,  j^Q.,, 
\<J^.  itc,  acconlini;  as  ilu-  jum  tion  ('  dixiilcs  the  pijK'  />/'  into 
two  portions  in  tlie  ratio  of  1  to  1 ,    1  to  j,   1  to  ^.  etc. 

Noll-.. —  riif  mori-  (.(irriit  v.ihu'  of  ,•■    is 


/..• 


/,    )■/     •    \ 


'  I  ■  ■  It  ■-• 


l:M 


.111(1  tli<    iii.iMniuii'  valiu    I'l      ,     '   '.    .,<iius  \]<>\   r\ni(l 

Oriji.)  hid  hy  livo  J\,.<!ir',oirs. — Neglect  all  losses  of  luaii 
t  xcept  the  losses  iliie  to  frictional  resistance. 

When  tile  valve  at  f'  is  closeil  the  flow  is  uin.llv  trom  ./ 
to  C,  and  the  ilciivery  is 


■J 

31,  »| 


\        ■       ''^f^ 


196  (miii'.F  hhn  HY  7H'i  rfsfki-iVrs. 

'I'lu'  line  of  ili,u;.;<'    lu'di  inlii"  i;r,idiciU    i^  .U\.   \\herc 


J/A' 


A, 


.\'/' 


i' 


'  I 


(^ji.  II  the  valve  a  little:  .1  vojimu'  (J.,  will  now  iV.w  thr'uij^'h 
< ',  .111(1  .1  vi>luir'-  ('.  into  (  ,  where 

'{"he  "line    itih.u^e       heeuiiu-i  t'"-  l>roken  hue  .I/l.\ 

\ -;   till'   openinj^    •'•    •'"       ''\'         ntniucs,  the   ])re->'<ure-h<a(l 

,/< 
ill    (■  <Miininslie>;.  anil    \\U'  ■■'  '       the  line  nf 

ih.irge  is  J/;!A'.  jAheinj;  hmi/.'iiia:  1  u  <'ii'^tatic  ecjiiilihrunii 
i'j  Minv  establishetl  hetween  ''  I'l.'  •  .nil  the  whole  ot"  the 
water  from  .7  pusses  thronyh  >  ry  bcinj(  tjiven  b\- 


C: 


\ 


Openmj;    ('    --iiii    I 
(iril'iee.  .uul  the  lino  "' 


I  ,    will    ■'11  \  I     the 


F.X.^Mri.E. 


107 


When  the  valvf  i-^  fiill  o]Hn  tlu-  ■•  liiu-  nf  i  h:ir..rc  '    i-^  M  y^  . 


where  },0  :=   ''',  .mii  thi    .h^Ji.in-, 


V  ,.  - 


Th 


le  siip[>i\'  Itoin 


IV  e<i.i,ii   li'  th.it 


\\  hen 


The  aho\e  investiL.aion  shtiu>-   tin    .uh  .int.i.^'e   ot   a  seoMul 
reservoir    in    enu'tf^eii?    cases    wlien    ,m    cNcessive     suppl)-    i- 


suiiiicnly  ilemamlc.'. 


.IS,  eu 


,  on  tile  (n  casii  m  nt  a  tiie 


Ex.  A  24-111.  pii'C'  .  //'  (icKAj  n.  K  11^.  ( .  iiiui  Is  IWD  rt-sfi  VDir.'',  tlu'  dil- 
Icrciiie  i>(  level  biiwi-iii  the  w^iii-rsur'-ucs  hfui«  ^so  It  From  a  jiiiu - 
li.'ii  ('  between-/  and  A' a  l-'-in  pijie.  C,  3000  ft.  loiip,  i<.niUHts  wil  li 
a.,  interiiicdiate  reservoir  liaviiii;  its  water-surface  ijofi  above  tli.it  (f 
the  lowest  reservoir.  Disius-;  the  HistiihiiiiiMi  (./)  when  AH  -\:.«i(i., 
ill)  wliiMi  ./('  4000  (t.;  ami  tinti  (. )  the  position  of  O  so  thai  ihr  c  ma" 
be  no  tluw  m  ('C. 


P" 

T 

/c 

1V 

_  i 

Fic    II!  F"-  "6' 

Take  the  Icwim  watrr-mrliue  n^  thnhitinn  plane.     .Mso  assume  tl'.it 

«»—■'—  .lX)02. 


1 

* 

't 

p 

u 

'' 

'  * 

i  fl 

m 

-M 

'-^H 

lr)S 


x  -t.vri  I . 


If  a    piivotiiflor   IS   inserted  ,it  i\  llic  wjter  will  rise  m  it  to  ,i  liefi-lit 
5  above  datmii.     Then 


(a)   Fig.   115: 

Between  .  /  ,111(1  0 

250-  - 

2000 

=    <T           =   <«7-,'. 

Hetwecn  t.)  and  /.' 

±JSo  I 
3000 

- 

ISetween  (  '  and  ( ' 

4000 

-..";' --'•^•3'. 

Il) 


T")  iiiul  the  lihectii'H  of  the  How  in  Cc'.  let  :  =  I ;o.  then  r',  =  o, 
"?'•'=  s',>-  '"■>'  =  »'6.  and  tlicrcforc  ?■,  >  7a.  Thu.s  moie  water  fl..ws 
Irom  ./  tu  C  than  is  requited  for  the  lowest  reservoir,  and  .i  portion 
must  flow  to  the  intermediate  reservoir.      Hence  z  >  150  ft.,  and 


<h 


(J.  +  'J'. 


7'l    =    —   +  7',. 

4 


Therefore 


/-ISO-.  ^   ,     /r-,50         /- 


t 


•m. 


<"■  ♦  '500  —  6r  =      «  2a  —  (GO  +  t  3.-. 

By  trial   this  yives   .-  =  i^i    ft  ,  very   nearly,    and   then,   siihstitutinc 
in  eqs.  (I). 

'!'>'  -  222,5,     ir     ■••  =  U'Jif^i  ft.  per  sec. 
«V  =  9l.  or    ?',  =    3.027  •• 

?•,*  -  201.25.  '"     ->  ^  14.186  •• 

Hence,  also. 

22     2' 
Q'  =  Y  -T  ^   i4-9'^»  =  46  879  cu.  ft.  per  sec. 

,.         22     I' 

O,    ==:      ^     .  -    X       3.027    ..       2.37S         •■ 
22      2' 

Q,^^.-^n'i^<;^  44.584     " 
and  Q,  +-  0,        4r).(/i2  =  Q,.  very  nearly. 


EXAMPLE. 


199 


(b>    Fig.  116: 

iJetwcen  A  ami  ' ' 

40.  X) 

Bftwecii  ''  .uiil  I' 

±_i5o  T 
3000 

r  -tw :  ' '  .iiiii  ( ■ 


KIT', 


ill) 


If  -  =  i;o.  r  .  =  o,  <rr.,'  =  ;,..  and  aW  =  A-  Tlu.s  r.  -  r.  and  there- 
fore r>,  ,.  ^',  ,  so  that  more  wale,  flows  to  the  lowest  reservo.r  than  is 
supphed  b?  the  highest  reservoir.  Hence  the  bahu>ce  n.ust  come  from 
ihe  intermediate  reservoir  and  :       i  ;o  ft. 

Also,  Q'  +  'J'  ==  '-'■• 


Therefore 


4000(i         4 


r  irw-wi/i  J.  (>000 


./' 


i    ly: 


i  ^oo 


I  62. 


„f,.  v.TV    iK-ailv.    .1.1.1    ili.-n,   Milistituting  in 


By  trial   this    ^ivrs 
r-^s.  ill), 

,,,'  =   192.5,    or     T'.         i3-'V4ft    ii.-r?e<.. 
?•,'  =  45.        or    J'j  =    6.709 
7.,'     -  ?4o.       or     T'i  =   15-49-  " 
Hence,  also. 


G.  = 


7  4 
2;  I 
7    '4' 

22       I 

7'  ■  4' 


1'  .  6.70<;  ■-     5=71       '■ 
2'  X  I5.4')2  =48.689      " 


=  G'  +  L'«  ^''■''v  '""^'■'y- 


(O   I.et  /4(>    -  X. 


XluMi   M.H.-  f,  =  o,  r  =  150  ft..  .""I  therefore 


250^^50  ^„.,,. 


150 

6000   —  A.' 


Hence 


fKXXJ  -  .1  ^  I  50  ^  3       a„,,      ,.  :=  ,400  ft. 
.«  100        2 


I'RCViir.M   Oh    THREE  RESEHfO/RS. 


m 


l'K"i'.MM    11.      C.ivcn    //,  ,    //,.    //,;     (J,,    ^),^,    and   thercfnn- 
L\i=   ±  (J,  +  Q,);   to  hiul  ;•  ,  ;^.  r,,   :•  ,  r'..  ;•,.   ~. 
A-  Iictorc,  let  J  be  tlic  prc-^surc-liuail  at  ( '.       'I'Iumi 


.      .      (i)      and      (_\  =   tt; 


//,  -  .r, 

7', 

-  ~ 

I 

7, 

= 

--  -  ^ 

«-■■• 

/, 


•     •     13.)       '■       Ll^  ''r/r/, 
■     ■     is)       •'       a.  =  W'V 


4) 


(6) 


1  luse  six  ctjuatioii-;  cmitain  tlic  seven  retjuired  quantities, 
\'i/-,  '\,  r,,  ;•,, ,  7'|  ,  :;,,  v^.  and  r.  'I'luis  a  seventh  eciuation 
must  '■;•  i.htaiiud  lurcire  tlien-  \alues  cm  1  ^e  tound,  '1  his 
<:<iuati()n  is  ^,nven  h\- the  condition  "  tliat  the  cost  of  the  pipin-; 
laid  in  place  should  be  a  inininuun."  it  beiiiL;  ,.  s,nne<l  that  the 
cost  ot"a  pipe  laid  in  place  is  proportion.  1  to  it:  diameter. 
I  leiK  e 


A'-,  +  ^.r,  +  /.'-..  -  •• 


ininimiun. 


J'Voiii  equations  (  i  ■  and  i  2 


(-;», 


«C^' 


^3)    ••    (4). 


.5j    ••    <<J|, 


±  -  Tj,  _  <>Q./ . 


-A. 


if  5" 


71 'r. 


Differentiating,'  these  three  equ.itions. 


/.-  .v;«  '^'.• 


(7/ 


Af.-//.V-S    UIIH    s/.*-' 


20I 


■Rut.  b\-  (■(iihitinii  17  I, 


/,,/;•,  ^-  /.,./' 


Ill 


^) 


V. 


hicli  is  the  seventh  cqu.itioi. 


This  hist  eij' 


1II    ''.-IV    IH 


the  K>rni' 


and 


=  ±;.  +  " 


Mains  with  any  Required  Number  cf  Branches, 


25- 

l.rt  th<T<    he  >i  junction-  ami  »/  PM"  - 

l^^t  u   .  /■'. ^'>,.  ''•-■  t'"'  "''   1'"  '^^""■ 

c;ich  successive  h'n;-;th  of  [hih-. 

11  L>--uri 


-hf.uis   at  tlie   end  of 


Let 


~ r„  l)e  the    1/   1 

3d,  .  .   .   '/th  jiuKti'in^ 


-lu  a.K    It   the    1st,   Jd, 


Let  /, 
I'KOHI 


.  /    lie  tin    Kn-th-^  of  the  I'l  I'M' 


IM    1       (iivm   /(. 


to 


I 


'Yhcrc  an-  w  .■(Hiati.  in^  d  tin   tvpe      —  ^^ 

Also,  the   .1     inlit\-  tlouin-    thron-li  tlu-  fir-t  ,.,.rtiMn 


.f  tin 


il  tc  the  Mini  "i  the    iiuantitu-  tlnumi;    thrcu-h  ail 


main  is  ecpi 

the  br.-.n  hesattlu-   fir-^t  juneiinn.  an.i   an    anah.-ous   .■, 


u.itum 


xvill  hold 


h  "I   tlie  reinainin 


1     lUIUtli'll'- 


Hi-^  // 


ditiiinal  ((luatii  n)'^  are 


)btained 


l'r>.ni     these     w  -\    n    e(|uations 


li'ini 


!    ana!>licull\-    ■  t    l'>     tlie    nu 


thcd    of 


rej'eated    ipj 


III  esiiil.itioll. 


202     y.-iRirnos  or  rri.oai  y  is  tk.-i.vsi'i-ks/-  s/.-v./vo.v. 


^-^ 


l'l<"lil  I  Mil.       (.ivcil  ./,  ,  //. //,„,    (J^,    (J_^  (J^    .   tf, 

'1  liurc  arc  imw   ,.iily  w  i^qnatioii-,  ,.t  tin-  t\j)i- 
±  //  T  =  :■■' 

I       =  "7- 

invoKin-  iii  -\- u  iinkiiMUTi  .iu,nititir>.  ;ui,i  tlu-   prohiciii   .uiinits 
ol  an  intlnitc  muiihc-i'  i  if  snliiti'>ii^, 

ll  1--  tlicrcfnrr  as-^nmcd  that  the  tost  of  tlu-  |)ipin<,r  laid  iti 
place  is  to  he  a  n/inniiiaii.  'lluis  ;/  ik-u  cciu.ition.  arc  obtained, 
and  tlic  „/ .-\  ,1  ccpiations  may  he  solved  analytical!)-  or  hy 
repeated  trial. 

\<.ir,_lh,.  niaxiniiini  velocity  ot  tlow  in  town  mains  is 
from  J  to  -  ft.   pcT  '-ii.ond. 

26.   Variation    of   Velocity  in    a    Transverse   Section.  - 
Assiniifn,'ii.~[h.i\  the  water  in  ,m_\-  portion  of  a  j)ipe  :s  made 
lip  of  an  infinite  inimher  of  li,,ll,,w  concentric 
<.\linders    of  tliiiil.    each     nio\  in.i;    [j.irallel    to 
the  a.vis  with  a  certain  detinite  velocity. 

I-et     i<    i)e    the    velocity    of  one     of  these 
c>linders    of    radius    .i     and     thickness     ,/x 
■i"  "    'll''    '!""     across    ,1    transverse   section 
Fig.  117.  '   is  L,M\en  In    the  ei|iiatioii 


(itj  —  2 /T.I   ,/. 


1   .  It, 


and  tlu   total  flow 


./  ll 


.') 


/•  beinj;  the  radius  of  the  i)ipe. 

If  .■•,.  he  the  mean  velocity  for  tile  u  hoj,-  transverse  secti.^n 
cf  the  pipe. 


?'-,= 


Q        2jl"x,/x 


nt- 


i2) 


i-.-tRifiios  oi    yfi.ociTY  /.v  TK.-issi''f-:Rsf-:  section.     203 

.\L;aiii.   ;is->uinin^    uith    N.uicr   tluit  the   surface   rcsist.mcc 
Wt\sccn  two  cuiucntric    .yluulcr.  1.  nC  tlu-    nature  nfa  viscous 

rcMstaiue  an.l  niav  l.e  represenle,!  l.y  /■'|'  per  unit  of  area  at 
the  ra.iius  a.  /  hem-  a  coefficient  called  the  coefficient  .4 
viscosity,  then  the  total  resistance  at  the  radius  .1  for  a  length 
(h  of  the  cylinder 

The  total  resistance  at  the  radius  x  -f-  ,{.i 


—  _)-  2  7r/{-  .  (/.v 


dx         d.\\    d\i        J 


Hence  the  l^t, 


li  re-~ii 


t.uit  re->i-,lance  tor 


the  len<'th  as  of  the 


LVlinder  under  consideration 


'1  he    conip'iiient    of  the  wei 
the  direction  of  the  axis 


/x\  dxj 
rht   of  the  slice  of  the  cylinder 


2nx  .  ( 


/.I  .  ds  .  sin  'K 


It 
I' 


*^  being  the  inclinati 


.f  tl 


ic  axr- 


Lj-t  _  ,/„-  l)c  the  fall  oi  le\el 


to  the  hori/on. 
n  the  distance  tis 


Then 


(/;;  =  r/>  .  ■-in 


'I  herefore  tornpoue 


lit  '«f  umnht  in  ill 


rectioii  of  axis 


=  —  Tf  .  :  T.r  r 


/r  .  r/,7. 


n% 


204     i'.tiu,-;iio\  oi-  n-.nH.tn 


ilOM. 


l~ 


The    resultant    prosurt-    nn    tlic    slui.    m    the    dircaion    of 

IlKitioIl 

=  !/—(/'+  ,//>i;2,Tt      ,ix  —   —  2n-.)   .  ,/.t      (//>. 
IliLMi,  siiuf  iht'  iiidtiiiii  is  uniform, 


and  therofiirc 

k  .  ds  d  I  du\  dp 

Inte^'ratini;  only  for  tin-  rN^iiukT  under  Cf)nsidcrati()!i, 

ft 

Rut   •' +  ,      '^^   i-Aid(  ntl\-  indi  iifiident  of  .r    and   is  a   linear 

futu-tion  of  .V     Alt     ;,  1  hap.    11 ,.      1  knee 

I      dj     'iU\ 

-  /.i-*^/  j  --  'I  totistant  =:  A,  suppose. 


rilcrrton- 


Integrating, 


d.xVdxl  =  ''''■ 


•     (3) 


(tl<  v 

dx  2 

Assumiiv^r   tint    the  i  entral  fluid    fil.iment  i-  tlie    filament  of 
maximum  veh.utv     Ih.  n  uli,  u  1    ^-  o,   '^^^^  is  also  nil.      1  herefore 

di,      .;,-' 


^  =  o,      and     .1 


dx 


y.-1KIMlli-)\    Oh    yLLOaiY   /A     IK-tSMIKil     ^H.IIUS.       205 
.iti'l  tlioroforc 

'^^a' (4) 

dx  2 


Iiitc'ratiiii;,   V.i\    4, 


4 


C  bcint,'  ;i  constant  of  intc^^ratinn. 

Since   '//>  is   the   difference    u{  intensity   nf  pressure   <.n    the 
ends  of  the  cyliiuh-ical  shce, 

-  271X  .  </.v  .  /•  ,     ^   Tx-  .  dp     :    nx".i'  .  dh. 


Tliercfor!. 


dx^  "   2I:  ds  ~  ~  "iX-  • 


,inii,  1)V'  eiiuation  (4), 


Let   /^„,,    be  'he    veh>cit>    of  tli.'    central   tllainent,  i.e.,  the 
value  ot  //  wlien  X    -  o.      'llun 


A 


"m«.  —    " 


III-, 


.         (5) 


where  P 


A 

4' 


Ayain.  l>>   e([iiation  (i), 


/)^' 


* 


aiu)  b)-  (.(iuatiiin  i2;, 


i^ERSE  SELTION. 


If//,  -  velocity  ..t  ,.i,H.  u.,11,  ,lK„.  In-  ,^,,,tinn  ^5). 

"'  =    "ma,    -   /V- 

Hence,  by  c.juations  (61  and  (7), 

"-  +  *.n.«.  ^    2r  „. ^gj 

If /.  =  o  when  .r  r-  ,..  tlu-n  T    .  _  ^''',    ,„„, 

4 

riKTcforc 


^^=  -'2   J    ^<'^-^)dx 


Anr' 


ivinr* 


If: 


tlu.    at   or.l.narv  velocities  of  imv   .?n,^',  "'■"'■•■*   •''''   ""«"•'• 

rut.,,  of  ,lH.   ,„.an  v,C,t  fr'n  '  """""'   '■"'"'"••"^  -1- 

tl.c-   n.,r,n..l  .  urve  ..f  vrlon,,,,    s    m  T""*'  cro,H.,ect.u„  ;  ,!,«, 

d.Htnrbance  is  f.-lt  m..nv    h      Ir,   ,  v      T'^  """    ""-'  *''^''^'  "'  "   ""- 

»'-  ';•'  "  "'- "'   "-^  iXXr'ir  a.  wl„el,  ,......, 

Sinootli  inter. Mr  -■•K"nirm     ,j    .,,    lictesUiiry    us   « 


IVATF.R-METF.RS. 


J07 


27.  Gauging  of  Pipe-flow.  A  variety  of  meters  have 
been  designed  t<>  register  the  (iiiaiilil\  ot"  water  delivercLl  b}-  a 
])i|)(.        'Ilu-  princijial  reijiiisites  of  sucli  a  meter  arc: 

1.  i  li.it  it  slioiild  rcgi-<ter  witli  acciirac\-  the  quantitN-  nf 
water  deli\ered  under  ililTereiit  pressures. 

2.  That  it  shiiuld  1)1  >l  ai)i>reciably  diini"i>^l'.  the  efTecti\e 
pressure  <>f  the  water. 

V  That  It  sliould  be  emiipaa  .iiid  adaptabli-  tn  every 
situation. 

,j       I'h.it  It  sliould  be  simj)le  and  dur.d)le. 

//u  Viiitnn  Milii-  fFig.  1  18)  is  so  called  from  X'entiiri. 
who  first  pointed  out  the  relation  between  the  pressures  and 
velocities  of  flow  in  converging  and  diverging  tubes. 


Ki.;.  11.1. 

As  shown  In  the  longitudinal  section,  I'ig.  1  19.  this  meter 
consists  of  two  truncated  cones  joinetl  at  the  sm.-'.llcst  sections 
by  .1  short  throat  i>icce.  At  J  and  /.' there  are  air-chambers 
with  hole'-  for  the  insertion  c'  piezometers,  by  which  the  fluid 
pressure  may  be  mcasure<l.  Hy  Art.  5,  (hap.  I.  the  theoretical 
tpiantity  Q  of  flow  through  the  throat  at  A  is 


',''. 


\  .i .    —  (I. 


»  2,4',//,  -//,). 


30S 


II'. ITER-METERS. 


rt, .  n,  bcin^  tlic  sectional  areas  at  .  /  an.i  /;.  respectively  ,  .ui.l 
//,  -  //,  the  difference  of  head  in  tiie  pie/.. .meters,  or  tiie 
'■  iiead  on  Ventiiri,  "  as  it  is  called. 


inlro.lMcin-  a  coefficient  ..f  discharge  T.  tiie  actual  clelivery 
thniLi'di  ./  is 


"1. 


•  ,v  -  ,/ -■ 


y^.iri/f,-  II, 


An  elaborate  series  of  experiments  hv  Herschel  -ave  C 
values  var>in-  between  ...a  and  1.04.  but  tl..-  ^reat  maioritv  of 
tile  value-^  la\-  between  .90  and  .()<,. 


h|i.    130.  — Sthonheyder  »  I'otiiiive 
Meter 


f"l>:.   t;i._Thr  I'nivrr!!,.! 
M.icr. 


I'll.'  pie/oinetM,    i,,„_v    Im    i,.„n.cted    with    a   recorder,  and 

tlms  .1  continuous  re-ister  of  tlie  .iuantit\-  of  water  passing 

tlirou^r],  the  meter  may  lie  oblaine.l  at  any  convenient  position 

>vitlnn  .1  r.ulins  of  low  ft.      This  distance  may  be  extrnded  to 

■\.  r.d  mil..  I,y  means  of  .m  electric  device. 


H'ArER-METERS. 


209 


( )tlRT  meters  may  be  ijeiKT.ill)-  classifitHl  as  I'iston  or 
Reciprocating'  MLtirs  ami  Inferential  or  Rotary  Meters.  Tliey 
are  all  provided  witli  reeorders  wiiich  re^Mster  the  deliver\-  with 
a  i^reater  or  less  deLjree  of  accuracy. 

llie  piston  meter  I'i;^.  I  20'  is  the  most  .ucurate  and  ijives 
a  positive  measurement  of  the  actual  dcliverv  of  water  as 
recorded  by  the  strokes  of  the  piston  in  a  cylinder  which  is 
filled  from  each  end  alternately.     Thus  an  additional  advantage 


Flc.  122.— The  liiilTalo  Moicr. 


Fl'i.  I2J.  — The  Union  Rotary 
I'istnfi  Mricr. 


IJossesscil   !)>•  a  water-eni;nie    1-  th.it   the  workiiij,^  c>lind.  r  ivill 
;dsii  serve  as  a  meter. 

In  inferential  meters  a  ilruin  or  turbine  is  actuated  b>  tlu 
f..rce  of  the  current  passing  through  the  pipe,  but  it  often 
happens  that  when  the  flow  is  small  the  force  is  insutHkient  to 
cause  the  turbine  to  revolve,  antl  there  is  conseciuently  no 
rej^'ister  of  the  corresiwnding  ijuantity  of  water  passint;  through 
the  meter. 


I 


!IO 


EXAMPLES. 


EXAMPLES. 


<  S.H.    Take^   -  32  and  (.i  g..llu„s  .-^  ,  cu.  ft.  unless  otherwise  specified.) 

1.  .\  wal(  ■•-tnain  i.s  to  be  laid  with  a  virtual  sloi.c  of  1  in  S50.  and  is 
to  give  a  inaxi.nurn  discharge  of  55  ruble  feet  per  second.  Determine 
the  requisite  diaineterol  pipeand  the  inaximuni  velocity,  t;.kinj.;/=.oo64. 

^Ins.  3.679  ft.;  3.28S8  ft.  per  sec. 

2.  I-ind  the  loss  of  head  due  to  friction  in  a  pipe;  diameter  of  i.ipe 
=  12  in.,  length  of  pipe  =  5280  ft.,  velocity  of  flow  -.  3  ft.  per  second  ; 

/  =  .0064.     Also  find  the  dis<-hargc. 

.his.  19.008  ft.;  2  35r,2  cu.  ft.  per  sec. 

3.  A  pipe  has  a  fall  of  10  ft.  ,„.,  mile;  it  is  10  mdes  long  and  4  ft.  in 
dlaiiKJrr.     I'iiid  the  diM-harge.  assuming/ z=  .0064. 

W«j.  54.7  cu.  ft.  per  sec. 

4.  A  pr.c  dis.  har^es  250  gallons  per  minute,  and  the  head  lost  in 
friction  IS  3  ft.  Find  approxiniaiely  th.^  head  l<  st  when  the  riiv  har-e 
i>  30.1  gallons  per  minute  ;  also  find  the  work  consumed  hy  fiiction  "in 
'"'^'^  '^^"i'''*-  •'''"■•  4.32  ft.:  7500  fl.-lbs.;    .2.9^0  ft.-ll,s. 

5    U  hat  is  the  mean  livdraullc  deplii  in  a  circular   pipe  when   the 
.   ,     iliamcter    , 
icight  ^ —  above  the  centre.' 


w.iter  risi's  to  the 


'US.        X  diaimter. 

0    .\  i:-  M,  h  pipe  has  a  slope  of  1  j  u-^-i  per  ,„ile;  lind  the  discharge. 
'/"-    -^SJ  yhts.  2. 11 8  cu.  ft.  per.  sec. 

7.    The  mean  velocity  of  n,,w  m  ..  24-in.  pipe  is  5  ft.  per  seond  •  find 
it.s  vir.ual  sl..pe./l)eing  .00^,4.  ^;,„    ,  j^^  j^ 

S.  Calculate  the  discharge  per  minute  from  a  24  in.  pipe  of  4cioo  ft 
length  under  a  head  of  Ko  ft..  usi,u.  .,  <  oefficient  suitable  for  a  clean  iron 

'"'*■  '■''•J- 34909  cu.  ft.  per  sec. 

•<j.  Mow  long  docs  11  i.,,v,  I,,  ,  i„,,i_v  ;,  ,jock  whose  depth  is  31  ft  6 
IMS.  nnd  which  has  a  horizontal  sectional  area  of  550,000  s(|,  ft  tlin)Ugh 
/«'0  7-f'.  «  ir<  ular  pija^s  50  ft.  long,  taking  into  account  resistance  .:t  en- 

"'T''"'-ri  ■     ,  ,  '"'"■'   2'4n>in.  6.sec. 

10.  I  he  virtual  slope  of  a  pi|«;  is  1  in  700;  the  delivery  is   iSo  cubic 
feet  Iter  minute,      ri.i-l  ihc  .li.iinete,  and  velocity  of  flow. 

-^«J.  1.26  ft.;   2.401  ft.  per  sec. 

11.  Uetermiiu    i,..    ,„ete,  .,1  a  clean   iton  pipe.  100  (ect  in  length 

which   III  to  deliver  .5  cu.  ft.  of  water  per  second  under  a  hea-l  ..f  1;  feet* 
Assume/ :^   ooT),  ,  ^T  ,      ' 

i-.    .\    reservoir   of    lu.ooo  s.|.   ft.   area  and    lor,  It.  deep  diMharges 


Fxt.Mrirs. 


21  I 


thmugh  a  j.ipe  24  ins.  in  diam.  and  2c>;<)  ft.  in  length.  Find  the  velocity 
..I  tlnv.  What  should  the  diani.  be  in  order  t'riat  the  reservoir  may  be 
emptied  ill  two  hours?     (/=.oo64.;     .Ins.  1  j.;;;  it.  per  sec;  4.0923  ft. 

13.   The  pressure  from  an  accumidator  at  the  entrance  of  a  </-in   pipe 
/.  ft.  lon^' is  1000  lbs.  per  sfj.  III.     If  .\' is  th.   total  U.l'.  available  at  the 


ow  that  tlie   11. 1'    .ibsorbed  in    friitional   resistance 


"(5-6o)-V- 


f  beinj; 


.(XjSi 


^^3 


lot,, I 


14.  Th<'  delivery  at  the  end  of  a  3-inch  pipe  is  11.06  111'.  1  li' 
ciTeclive  head  at  the  cntraiue  to  pipe  is  896  feet.  The  loss  in  frutional 
resistance  is  21  percent.  Find  the  distance  to  which  the  rncr^y  is 
transmitted.  ■■!'"■  \  3.000  h.. /  \K\nfr  .00(^4. 

15.  A  reservoir  has  a  superficial  area  of  12,000  ft.  and  a  depth  of  f'O 
ft.;  it  is  emptied  in  60  niiniites  ihnmnU  /our  horizontal  circular  pipes, 
.•iHi:d  ill  dianieier  and  30  ft.  loiijj.      Find  the  di.iineter.     (/ =  .00^14.) 

Alls.  1 .78O  ft. 
ICxplaiii  how  the  f,>/,i/  head  is  n-ade  up,  and  draw  llie  pi, me  of  charge. 

16.  .\  3-incii  pipe  is  very  ^jradii, illy  lediiced  to  J  inch.  If  the  pres- 
sure-head ii.  the  pipe  is  40  ft.,  liiid  the  greatest  velocity  with  wliicli  the 
water  can  How  through.  --/'"•  1-4  ft.  per  sec. 

i;.  Water  flows  throiK^di  a  24-inch  pipe  5000  yards  in  leiii,'th.  At  Io<x> 
yards  it  yields  up  300  cubic  feet  per  minute  to  a  br.uich.  .\\.  2800  yards 
it  vields  up  400  cubic  feet  per  minute  to  ;i  second  br.mch.  At  4000 
yards  It  yields  up 600  cubic  feet  per  minute  to  a  third  biancli.  Tlic  cie- 
livcrv  at  the  end  is  500  cubii  feet  per  ininute.  Find  the  head  absorbed 
by  friction.     (/  =  .0075.1  •''"•  '~7-^'^^  ''• 

18.  Find  the  IM*.  rc(iuireil  to  i.u^e  5joi;,illons  per  minute  to  a  heijjht 
of  60  feet,  throii^li  a  pi|)e  100  feel  in  leii(;lh  and  6  in.  in  di.iineter,  the 
coeiricient  of  friction  beini;  .oo<i4.  Aiis.  10.74. 

19.  What  head  of  water  is  recpiire.l  f.ir  a  sin.  pipe,  150  ft.  in  lenmh, 
to  carry  oil  25  cu.  It.  of  water  per  minute  ?  Ahs.  1.56223  ft. 

What  heail  will  be  refpiiri-d  if  the  pipe  contains  two  rectangular 
k„ees?  /«.v.  t.84918  ft. 

20.  Determini'  the  <leli\.i\  ..1  ,1  2-in.  pipe,  4.  It.  i  ;iig,  under  a  5-ft. 
head,  f  being  .(X>5.  ••'»«•'■    '44')  <"•  ft-  pc*  »cc. 

What  will  be  the  delivery  if  the  pipe  has  5  small  curves  of  go'  rur- 
v.ituie,  the  ratio  of  the  r,i(iiuH  of  the  pipe  to  that  of  the  curves  being 
I  .  2  >  .Ins.  .1381  cu.  ft.  JHT  sec. 

21.  The  curved  buckets  of  a  turbine  form  channels  12  in.  long,  2  in. 
wide,  and  2  in.  deep;  the  mean  radius  of  curviiture  of  the  axis  is  8  in.; 
I  lie  water  flows  along  the  channel  with  a  velocity  of  50  ft,  (ler  minute. 
Wh,it  is  the  head  lost  through  curvature?  Alts.  ,00138  ft. 

22.  Find  the  fKiwcr  transnutteii  by  water  flowing  at  80  ft.  per  sec.  in 
a  36-in(  h  pip-,  the  metal  being  1)  inches  thick  and  the  allowable  stress 


U- 


213 


I-XAMPI.FS. 


t 


2800  lbs.  i.crsquar.'  iiuli.     If  x\u:  p,;,e  is  i^  miles  in  l,MiL;tii.  l"in<i  the  loss 
"'  '"'"■'''■•  ■■/'"•  576  H.P.;  7:.o.2  tt.-lbs. 

23.  Fitul  the  (liamcter  .,f  a  pipe  J  ,„ile  lout;  to  delivor  1 500  gallons  of 
wat.r  p.  r  iiuiiuie  witii  a  loss  ol  20  feet  of  heuil.     (/  -  .005.) 

■  1>ii-  1.0135  ft. 

24.  Water  IS  to  be  raised  20  ft.  tlinnigh  a  30-fi.  pipe  ol  6  iti.  (iiameter. 
Find  tin-  velocity  of  flow,  assumnii;  that  10  per  rent  of  additional  p.  .,er 
is  require<i  t  .  overcome  friction,  and  that/  =  .0075. 

Ans.  8.44  ft.  per  sec. 

25.  In  a  pipe  32H0  ft.  in  le.iu'th  and  delivering  6750  j,ral|ons  per  miii 
the  loss  of  head  in  friction  is  83  !t.    Takinj;/  -^  .0064.  find  the  diameter'. 

.Ins.  1.527  ft. 

26.  Calculate,  by  Thrupp's  formiil.i.  the  11  ,w  thrr)nj,'h  a  4-in.  roui-h 
wrouj4htiron  pipe  having  .1  fall  ..f  35  feet  per  mile. 

Ans.   .J426CH.  ft.  ]ier  sec. 

27.  A  cle.in  6-in.  pipe  has  a  viriu..l  slope  of  i  per  400.  lakinj; 
/  =  .005.  find  the  velocity  of  steady  How.  the  discharge,  and  the  ener^;v 
absorbed  in  friciional  resistance  in  1000  feet. 

Ans.  2  ft.  per  s.>c.;  l\  cu.ft.  per  sec;  6i,<,\  ft. -lbs. 

2>s.  .\  6-in.  pipe.  300  ft.  I.HiK,  dLscharne.;  into  a  3.in.  pipe,  also  5,K3fr. 
loiiy;.  'Ilie  elf.ciive  head  between  the  inlet  and  outlet  is  10  feet.  Find 
tlie  dischartre.  takini,'  /"=  .oo()4.  an.l  makin-  allowance  for  the  resistance 
"^  ^'»-' '"'^■'-  Ans.  .1703  cu    ft.  p.rsec. 

i').  How  (ar  (an  Kxj  II. 1'.  be  traiisnii'ted  by  a  3i-in.  pipe  with  a  loss 
of  head  not  ex<  eediiig  25  jier  cent  unrler  an  eifective  hea<l  of  750  lbs   per 

'^l"''"- ''" '•  ■'  .///..  5426.3  ft. 

30.  .\  pip.-  20.W  ft.  Ion;,'  .md  2  ft.  in  <!ianieter  discliarj^cs  at  the  rate 
of  |(.  ft.  per  second.  Find  the  increase  in  the  discliarp-  if  f,,r  the  last 
1000  ft.  .1  second  pipe  of  same  size  be  laid  by  the  si<le  of  the  first  and 
connected  with  it  .m,  th.it  the  wai.r  may  liow  erpially  well  alonfj  either 
I"'""'  Ans.  7.24  cu.  ft  per  sec. 

31.  .\  pipe  of  lenRth  /and  radius /- j^ives  a  discharRe  Q.  How  will 
the  di.scharKc  be  alTect<<i  in  by  <loublinu  fx-  radius  Un  the  whole 
length;  (21  by  doubling'  the  radius  .'or  half  tin  length;  (3)  by  dividing;  it 

into  three  sections  of  ecpial   lennih.  0!  which  the  radii  arc  r.  '-,  and  - 

'  2  '  4  ■ 

respectively  ?     (/  ~  coefficient  of  friction.) 

Ans.    I.   New  discharge  =  dOP*^  "*"  ^^^\^- 

-^.r  +"33///  ■ 


~VS^4.7i2r  + 


.  ,.-, ,   4J28//y 

32.  A  24-in.  pipe  2000  ft.  lonR  gives  .1  discharge  of  (J  cubic  "feet  of 
water  per  mmutc.  Determine  the  thanKC  in  Qhylh^  substitution  for 
the  foreKOin^;.. I  either  of  the  following  systems:  {I)lwo  lengths   e.u  h 


/;  \  nirt/iS 


213 


<>t  1000  It.,  wbcisc  (ii.imetiTS  aic  24  ins.  atid  48  ins.  respect ivelv  ;  i;i  four 
Ifiii^th?,  each  of  500  ft.,  uhost-  (li.niu-tets  .itc  24  ins.,  iS  nis  .  16  ins.,  and 
24  ins. 

Dr.uv  tlif  ■■  jil.inc  nf  cliarge  '  in  cai  h  case, 

/^'/.v.  fi)  Disci,  irj,'e    is   increased   332   per    cent  taking  loss  at 
char-j.;e  of  section  into  account , 
Discharge  is  increased  35,7  per  cent  disregarding  loss 
at  change  of  section. 
(21  Discharge    is   diminished    4s    per   cent    disregarding 
l')sses  at  change  of  section 
33.   (2  is  the  discharge  from  a  pipe  of  length  /  and   radius  r  .  examine 
the  effect  upon  'J  of  increasing  /  to  iir  (ui  ;i  length  w/ ol  the  pipe. 


yl>i.s.   New  tlischarge  =  (J 


2         r 


»i  f 


'tn\       {„*  —  I)'*  [■  • 


34.  A  5-in,  pipe,  3ix>  ft.  long,  discharges  into  a  3-in.  pipe,  200  't 
long,  llie  l..lal  fall  beiii-  5  feet.  Find  the  <|  i.aitilv  of  flow  in  g  dioii-, 
l>erhour.  .,,„    ^..x,, 

35.  A  main,  1000  ft.  long  and  with  a  f.iil  n\  5  ft.,  discharges  in'u  two 
branches,  the  one  750  ft  long  with  a  fall  of  3  ft.,  the  other  250  ft  long 
with  a  fall  of  i  ft.  The  longer  branch  passes  twice  as  much  water  as 
tiie  other  and  the  total  delivery  's  47!  cu,  ft  i.er  minute.  The  velocity 
of  flow  in  the  main  is  2'  ft.  per  second  F-nd  the  iMmelersof  the  main 
and  hranches.     (/=.oo64.)  ^//y.   (.3245  ft.      51ft.  ;.  36  ft. 

36.  The  w.iter  in  .1  12-in.  main  800  ft.  long,  flows  at  the  rate  of  1  ft. 
per  second  and  om-  third  ni  tl:-  water  's  disrh,irt;rd  intja  branch  zoo  It. 
long  with  a  fall  of  1  in  40.  while  the  remainder  passes  into  a  600-ft. 
branch  with  a  fall  of  1  in  60.  The  ellective  head  between  the  inlet  and 
o.iilet  of  the  main  is  2.1  ft.     Find  the  total  discharge  and  the  diameters 

f  the  branches,  taking  f  =  .0064.  and  making  allowance  for  loss  at  inlet 
but  disregarding  losses  at  the  uinction. 

Ans.  94J  cu   ft.  per  sec,  .27  ft.,  .39  ft. 

37.  If  a  pipe  whose  diameter  is  8  ins.  siiddenlv  enlarges  to  one  whose 
diameter  is  12  ins,  t'ind  the  power  required  to  force  1000  gallons  |)er 
minute  through  the  enlargement,  and  draw  to  scale  the  pl.ine  of  charge. 

Atfi.    Knergv  expended  =    1377  M.l*, 

38.  1000  gallons   per  minute  are   forced   through  a  system  of  pipi 
-•//>',  A'C.  tVJ  of  whiih  the   engtlis  are  100  ft.,  50  ft.,  and  120  ft.,  and  t'.e 
radii  6  ins..  3  ins..  ,ind  4  uis  .  respcc  tivelv      I'-.hv  to  ^,  ,iie  the  pianc  of 
charge. 

.7«j,"Los8  in  In. 


Iroiii  A  \\<  It  = 

A  10  C  - 

•■       tto/'  = 


14.744  It  ,   ioss  .11  /,'  .-  i4.5f,  (t. 
!.15')      C=    S.819" 

3436   •■ 


ai4 


I.X.-tMPLES. 


3>y.  A  |ii|)c  4  iiw.  Ill  (liaiiK-tLT  suddenly  contracts  to  one  3  ins.  i:i 
dianieu-r;  linil  the  power  necessary  'o  force  250  i^allons  per  ininulc 
liiroii^li  the  sudden  conlr.ictinii.  .tiis.  1.^3097  il.l'. 

40.  Wiiitr  tlows  troMi  a  j-in.  pipe  lliroM<;li  a  li-m.  urilicc  in  a  di.i- 
pimiffni  inlM  a  J  in.  pipe.  What  head  is  i.cpincd  il  the  dehvery  is  to  he 
¥,  cu.  (t.  of  wall  r  per  nunnle.-  .Ins.  2.S26  ft. 

41.  500  gallons  of  walei  pir  iniiMiti-  ate  forced  llirouf-li  a  c<ailit)uous 
hni-  of  pipes  -//.',  I!C.  CD.  of  w  Im  li  the  radii  .ire  3  ins.,  4  ins..  2  ins.,  and 
tile  lengtlis  loo  ft.,  150  fi..  and  So  ft.,  respectively.  Tiiid  the  /,>/,»/ I. iss 
of  liead  (<i)  ('iiie  to  tiie  .siiddc-ii  i  haiis^es  of  f.irrii  at  A'  and  ( ',  (/')  due  to 
friction.  Find  (,  »  tlic  di.iineter  of  an  e<)nivalent  uniform  pipe  of  the 
same  total  leiij^th. 

Am.  (ii)  .1378  ft. :  1.152  ft. 

ti)  3.6.S8  ft.  m  .//.';   1. 313  ft.  in  /)'c';  22.393  ft.  in  t/>. 
if)  ,4212  ft. 

42.  A/i,  JiC,  lJ)  is  a  system  of  tlireo  pipes  i.f  wiiich  the  lengths  arc 
lOJO  ft.,  50  ft.,  and  boo  (t..  and  the  diameters  24  ms..  12  Ins.,  and  24  ins., 
resjiectively  ;  the  water  llo«s  from  (  7>  throutjli  a  1 -in.  orifice  in  a  thin 
diaphragm,  and  the  velocity  of  flow  in  .//.'is  2  ft.  per  second.  I>raw 
the  plane  of  i  liarge  and  tind  the  mechanical  ellect  of  the  iflinx, 
/  being  .U064. 

.///.».  Loss  at  C -z  ,"„  ft.;  at  A' =  .;,/,,  ft.;  in  fnclioii  from  ./  to 
/;  =  .8  ft.  ;  from  /.'  to  t  ^  1.28  ft.  ;  ir<im  C  to  I)  -  .64  ft.  ;  energy 
of  jet  -r--  14..S11;  Il.l'. 

43.  1000  g.illoiis  per  ininiite  Hows  through  a  sudden  contraction  from 
12  ins.  to  8  ins.  at  ./.  then  through  a  sudden  enlargement  from  8  ins.  t<j 
i;  ins.  at  li,  the  intermediate  |)ii>e  AH  being  100  ft.  long.  Draw  the 
plane  of  cli.irge.y  being  .0064. 

Am.  Loss  at  -•/  =  .2S,S  ft.;  at  A'  -  .281  ft.  ;  in  friction  fonn  / 
In  A'  —  3.499  ft. 

44.  W.iter   Hows  from   uiie  tube    into  another  of  /j<'/i-c  the  dianuiei 
the  velocity  in  the  latter  is   10  ft.     Find  the  head  corresponding  ti>  the 
resistance.  Ana.  14.062;  ft. 

45.  .V  2-in.  pipe  ,/  suddenly  enlarges  to  a  3-in.  pipe  A',  tin-  f|uai,tity 
nf  waler  flowing  through  being  loo  g.illons  per  minute.  Find  the  loss 
of  Head  aii.i  'he  ddTereme  of  pressure  in  tlic  pipc-s  (i)  wIumi  the  (I..w  is 
from  ./  to  ,'.  .  I  2)  wiien  the  llow  is  from  A' to  A.  C.  being  .60. 

.Ins.  Ill  Loss 'if  head  —     .S.oji^  in. 

(lam  of  |ircssure-heail  =13.83      " 

(2)  Lo.is  111  head  —    7.42S   " 

Diminution  of  pressuie-head  =  29.88     '• 

46.  A  jin.  horizontal  pipe  rapiilly  contracts  l.j  a  i-in.  mouthpiece, 
whence  the  water  emerges  into  the  air,  the  ilischarge  being  660  Ibt.  per 
miniits.     I"inil  thi'  pressure  in  the  jiii.  tnaiii. 


J 


h:.\,-iMri.FS. 


21  : 


If  the  5  ill.  pipe  is  joo  ft.  m  It-iij^ili  iiii'l  rect-ivcs  water  fnnii  ,m  dikmi 
tank,  tind  tin-  tii-ii,'lit  >>f  ilu-  tank,/ lii-iiif;  .005. 

.Ins.   1003  5  1I)S.  per  -^(i.  ft.  ;    19.9-  It. 

47.  A  liori/onta!  pipe  -(.  ins.  in  (li.imeter  sii(lii<'niy  enhirnes  tn  a 
(iiaineter  of  (>  ins.;  linil  tiic  force  rc(juire'i  to  caU'ie  a  flow  of  joog.. lions 
of  water  per  niiiiuie  tlironj^li  the  sudden  enhiri;i-nii-nt. 

.///v.     OM  11. 1'. 

48.  1000  g.dlons  per  minute  is  to  he  forced  thront;h  a  ^y.^teni  of 
pipes  .//>'.  /'C.  C/>,  of  whi<ii  the  len^'.lis  ar  •  luo  It..  50  fi..  lao  ft.,  and 
tlif  radii  4  ins  ,  6  ins.,  atul  3  ins.,  respectively.  What  must  lie  diameter 
of  equivalent  uniform  pipe  }     l>r.iw  the  plane  of  rhar^'e,/  heing  .0064. 

Ans.  Diameter  =  3.4  ins.; 
loss  in  friction  from  ./  to  A'  =  11  !.</)    ft.;  loss  al  /.'  =     4.499  ft.; 

••      y."  to  (_■  =       7.372 C^     14.36    ■' 

"      C"lo  J>=  566.i7      ■■ 

49.  Find  the  H.I',  required  to  pamp  1,000.000  f^allons  of  water  per 
day  •  f  24  hours  to  a  liei);ht  of  300  it.  through  a  line  of  sirai^lit  pi|>inj,' 
3000  ft.  lon^',  the  diameter  of  tin-  pi"e  heini;  >S  ins.  for  the  first  1000  ft.. 
6  ins.  for  the  second,  and  4  ms.  for  tin-  third,  allowance  heinj;  made  lor 
the  loss  at  inlet  and  the  losses  .it  ahiupl  chanj;es  of  section  ;  also  4  is  to 
l)e  taken  as  the  coetricienl  of  resistance  for  pump-valves.  (Atcii.inyes 
of  section  t\.  —  .64.)  What  is  tin-  itiamcter  of  an  equivalent  niiilorm 
].il>e?  (/  =  .0064.1  yhis.   196;  di.im.  =  .403  ft.,  or  say  ;  in-. 

50.  In  a  Riven  lenulii  /of  a  circular  pipe  whose  inner  radius  is  r  and 
thickness  ^,  a  column  of  water  flowing  with  a  velocity  v  is  suddenly 
cliecked  hy  the  shnttini;  olT  of  cocks,  etc.     Show  tliat 


r/i  = 


/.VA» 


I    + 


/■:\     r 


c-c)efr:cient  of  elasticity, 
I  \-tcnsion  of  pipe  cir- 


in  which  //  =  head  due  to  the  vel<Mity  v,  E  - 
Al  —  coellicienl  f)f  compressibility  ol  w.iter.  \, 
cumference  due  to  li. 

51.  Tne  wall  r  surf.icc  in  one  reservoir  is  ;oo  ft.  ahove  datum,  and  is 
100  ft.  above  the  surface  of  the  water  ir  a  secimd  reservoir  2o.<x>j  ft. 
awav,and  connected  with  the  lir-t  by  an  iH-iii.  main.  Finil  the  delivery 
per  seconil,  taking;  into  account  the  loss  of  head  al  the  entrance. 

Alts.  7.64  cu.  ft.  per  sec../ being  .00^.4. 

52.  Determine  the  disi  h  irge  from  a  pipi'  of  1:  in.  radius  and  32S0  it. 
in  length  wliicii  connects  two  reservoiis  having  a  dillerence  of  level  of 
12S  It.  Take  into  account  resistance  al  entr;ince.  Draw  the  plane  of 
charge.     (/  =  .005.)  Ans.  4^.57"  <-'"•  't.  P"  sec. 

53.  Determine  the  diameter  of  a  clean  iron  pipe  5000  ft.  in  length 
which  connects  two  reservoirs  having  a  total  head  of  40  ft.  and  dis- 
charges into  the  lowir  al  the  rate  of  20  cu.  ft.  per  second.  Draw  to 
scale  liie  I'lie  .  f  charge.      (  f  =  .005  I  -Ins.  1.9219  ft. 


I 


?!S^! 


2X6 


F.X.4MPLES. 


It 


,4.    The  .i.tleronre  of  k-velbcuveen   tl,c  two  reservoirs  is  .00  ft    and 
l..y  .r.-  ,  o„,K..:lc,l   l,v  .  p,p.  ,0.000  ft.  lung.     Find  tl.e  diatnoter  o'f  t   e 
PM-e  so  aslo«,vead.scharK'o  of  .ouoc.b.c  fcc-t  per  minute  (,„  by  DarcVs 
f-nnl,.,  ,^,  assunnng/  =  .0004.     .Allow  for  loss  of  head  at  L.t'ra.Ke? 
...  -  ''"•  '•'>  --'56  ft,  if  ,r  =  .0001O22  :  (/,)  .  360  ft 

,,.  Uvo  rescrvous  are  connect.-.l  by  a  i^-incl.  ,,ipe  i4  uuWi  1,,,',,, 
lor  the  rst  500  yards  u  has  a  slope  of  .  in  30.  for  tl.e  nA  ,  „  ^^i 
.-lo,.e.,f  ,  ,„  .00,  :,„d  for  the  re.nain,!..,   .,f  its  len.yth   u  is  level.     The 


ile,,d 


lr!r  Ta  ""  "■'"'  '^  55  ft.  and  that  ov.^  the  outlet  ,s 


1;  ft. 


...  --        ji  .........   i.i.n  uvi:i    uie  out  et  IS  1 

Determine  the  discharge  i„  jrallons  per  mmute.     (Take/  =  .0064,, 

-/     T  4-4/is.  1050.66. 

/..    l«o   reservoirs  are  ,  onnected  by  a  6-i„.  pip,,  j,,  „„,.^.  ^^^.^ 
earh  se,:t.;o„  l,..,„.  three  .piarters  of  a  mile   in  le,.gth.       The  head  Zlr 
the  .nle,  ,s  .0  ft.,  that  over  the  outlet  9  ft.    The  virtual  sl„p.^    "  [."s 
sea,o,,  .  ,  ,n  50,  of  the  .econ.l  .  ,n  ,00.  and  .  .    -hird  se  tion        ij  d 
h.nd  <he  veloruy  u{  ll,nv.  .,nd  the  delivery./ being  .005. 

„     ,,.       .      .,      ,      ■'";;•  '*-5  ft.  per  sec;  3;,2  gallons  per  minute. 
5,.  A  pipe  ,  miles  long.  „f  uniform  diameter  e.pial  to    '2  in     ro„vevs 
w-.t.-r  from  a  reservoir  in  whnh  the  water  stands  at  a  lieiglitof  300   t 

h       ,7   r  '"«";^^"''^^''  '"■'^^-  '"  ''  ^-'^^voir  in  whirl,  the  water  st^,  is 
.  t  a  height  of  ,50  ft.  above  the  same  datum.     To  what  height  will  ' 

n^.   m   a  su,,ply-p.pe  taken  one  mile  from  the  lower  end  >     For  wl  . 
pressiire  u,,u   I  you  design  the  main  at  this  p..int.  if  it  lies  .0  ^    ,       ^ 
tllc  lev  el  uf  the  lower  reservoir  ?     t  f  ~  .0064.) 

cS     \     I     ,    ,  ■'"■'•  '"'>''55  ft:   19.13  Ihs.  per  s(i.  in. 

5-V  ■\  '  l.an  /,.in.  pipe.  ,000  It.  long,  has  four  sharp  knees,  vi/  one 
-f  6.^  .  two  of  <^"  and  one  of  ,..,'.  lind  the  head  wa  ted  at  he  k  n  •  s 
....i  ...  the  straight  pipe,  tiie  „ow  being  at  the  rate  of  .50  g.d';::;';:; 
\       •  '-I'li.   .J734  ft  •   3  o""??  ft 

59.   A  f.-.n.  pipe.  4.300  feet  m   length,  lea.is  from  a  reservoir  •    t,",    . 

k..gth.   the  one  leading  to  a  reservoir  //,  the  other  to  a  reservoir  c 

'"'■-":""■.''' water  in./;     ,1..  fee,   above  that   in  A  a  1      ,0.^^  f,  .; 

abov..  that   i„   c.     Fin.l  the   velocities  of  flow    in  the  three    br:"   Vs^ 

7:  p  '"I";  --^  '"•  '■"  'l'a.iuMerand  2ouo  ft.  long  leads  from  a  reservoir 
...  whie  1  the  evel  of  the  Water  is  4...  ft.  above  d.itun,  to  L  ;:  ' 
«  h  ,t  divides  into  two  branches,  vi...  a  .>in.  pipe  /.'C.  .ocl.  f,  L^ 
le.  dmg  to  a  reservoir  m  which  the  surface  of  tl.e  water  is  .50  feet  at,  ve' 
atum  and  a  branch  /;/;.  .,,0  ft.  long,  leading  to  a  reservoir  i n  h ich 
le  surface  of  the  water  is  50  f,.  above  .la.i.m.  |,.,..rn,i,ie  the  dian  te 
of  /-.  >  when  the  free  surface-level  at  /i  is  „„  300  ft.;  ,/„  2:0  ft  a."  1  f^> 
.00  above  datun,.  ..,„,  , ,.  ,,5,  f,^  ^  ,^^^\l^  =^^^^<^  <0 

61  Two  reservoirs  ^  and  //  are  connected  bv  a  line  of  „i„inri/(M- 
2000  ft.  m  ength.  From  the  middle  point  r>  ..f'this  p.pe  a'  Kcl  /V 
.00.  f,.  1.,  length,  leads  to  a  reservoir  C^     The  reseills. ./;;;;"  V'r,: 


i;X.-)MPLES 


20O  feet  aiui  ickj  feet,  respectively,  above  the  level  of  C      Ttie  deliveries 
ill  AfO,  OP.  0.\:  in  cubic  feet  per  second,  are  V.  \'t.  and  n  respei  iivlv. 
Find  Ut)  the  velocities  of  flow  in  MO,   OP.  0\:  (/?)  the  radii  of  these 
lengths;  i.)  the  height  of  the    free  surfact-level  at  O  above  C',/ being 
.00(4.        .ins.  (a)   1 1. 1  21   ft.  per  sec.  in  Ml)-.   10.158  ft.  per  sec',  in  (V: 
14.145  ft.  per  .sec.  in  O.X. 
(/')  .5  ft.;  .4183'  ft:--f'5«8ft.         (<>   150.5  ft.,  very  nearlv. 
6>.   rind  the  amount  of  water  in  gallons  per  day  wliicli  will   be  de- 
livered by  a  24-inch  cast-iron   pipe,  15,000  ft.  in   total   length,  wiien   the 
water  surface  at  the  outlet  is  Kji  ft.  below  the  water  surface  at  the  inj.-t, 
t. iking/  =  ,001  and  allowing  for  resistance  at  inlet. 

If  tlie  water,  instead  of  flowing  into  a  reservoir,  is  made  to  drive  a 
reaction  turbine,  what  must  be  the  velocity  of  flow  in  the  pipe  10  give  a 
max.  speed?  Wluit  will  be  the  H.P.  of  the  turbine  if  its  efliciency  is. 84.' 
.\  third  reservoir  i^-  connected  with  the  system  by  means  of  a  24-iii. 
cast-ron  pipe.  7500  ft.  long,  joined  to  the  main  at  the  middle  point. 
The  water  surface  of  this  intermefliate  reservoir  is  50  ft.  above  that  of 
the  |.)west  reservoir.     Discuss  the  distribution. 

.his.     22,628.571?;    7.7   ft.  iierscc;    12.63   H.P.;   r  =:  73.6S  or 

51.33  ft.;  fi  =  7.76  or  12.42  ft.  per  sec;  ',;  ~  10.05  or  2.373  fi.  per 

sec;  T,  =  17.73  or  14.8  ft.  p-r  sec. 

63.  The  water-levels  in  two  reservoirs  J  and  /.'  ;ire.  respectively,  300 

ft.  and  200  ft.  above   that   in   f.      The   leservoir  ,-/   supplies  3  cu.  ft.  of 

water,  of  which  2  <  u.  ft.  go  to  /,'  and  I  cu.  ft.  goes  I.)  f.    A  pipe  2500  ft. 

long  leads  from  .  /  to  a  jiiiu  tion  at  O.  from  which  two  branches,  eai  h  2500 

ft.  in  length,  lead,  th^;  one  to  //  and  the  f.tlier  to  C.     .Assuming  that  the 

cost  of  laying  a  (lipe  in  place  is  proportional  to  the  diam.  and  that  this 

cost  IS  t.)  be  a  niinimun,.  find  the  pressure  head  at  ('  and   the  diirns.  of 

the  pipes. 

.Ins.  164  ft.;  diain.  of  .10  =  .66  ft.,  of  OP  -  63  ft.,  of  OC  =  .4  ft. 
'>4.  An  engine  pumps  a  volume  of  (_>  cubic  feet  of  water  per  second 
through  a  hose  1  ft.  in  length,  and  </  feet  in  di.imeter.  having  .it  the 
end  a  nozzle  J)  feet  in  diameter.  Find  the  pumping  U.W  and  apply 
your  result  to  the  determination  of  the  Il.l».  of  an  engine  which  is  to 
pump  30  cu.  ft.  of  water  per  minute  through  a  i-in.  nozzle  at  the  end  of 
a  3-in.  hose  400  ft.  in  length  (/=  .00625).  Also  find  the  fori  e  r.<|uiied 
to  hold  the  nozzle.  ,;,/,.  ,  ,,'j  U.P. ;  8<jJij  lbs. 

65.  A  lire-engine  pumps  water  through  a  400-ft.  it'ngth  of  2{-in.  bore 
at  the  rate  of  12  ft.  per  second,  and  discharges  through  a  i-in.  nozzle. 
Find  the  pressure  in  the  hose,  and  the  pumping  FI  I>.  Also  find  tlu- 
ffjrcp  required  to  hold  the  nozzle.     (/=  .00125.) 

.hts.  .6702  Ihv    per  ft.;   5.0.^16 ;   59.95  |bs. 

66.  The  roiidiiit-pi[ie  for  a  fountain  is  250  ft.  long  and  2  in.  in  diam- 
eter ;  the  (  oellli  lent  of  resistance  for  the  mout'ipiece  is  .32;  the  entrance 
orifire  is  sulficieiitlv  rounded   .nu\   ij,,-  bends  have  suffieiuiitiv  long  radii 


jkS 


tiXAMri.rs. 


■  il  ciirvauiro  tn  allow  if  ilii-  corrcspondiiif,' coctriciint  of  resistance  hemp 
(lisi(i;ai(le<l.      llow  liiyli  will  a  1-iii.  jet  ri>e  imilfr  a  iiead  of  30  ft.  - 

.Ins.  20.4  ft. 

67.  Water  siitfa<(;  of  a  reservoir  is  300  ft.  .ibovc  fiatiiiii,  and  u  4-111. 
pipe  600  ft.  lonj^  l.-aiis  from  reservoir  to  a  point  200  ft.  above  datum. 
Find  liie  lieii^iit  to  wliirli  tlie  water  would  rise  (,j)  if  end  of  pipe  is  open 
to  atiiiosp.iere.  ((5)  if  il  terminates  in  a  i-in.  nozzle.  In  latter  case  find 
!ouj;iiiidiiial  force  on  nozzle.        . ///,v.  (,/)  2-  It.  ;   yfi)  87.52  ft.;   H).(:()i  lbs. 

68.  I'lie  surface  i.f  the  Water  in  a  tank  is  388  ft.  abf.ve  datum  and  is 
conn<-cted  by  a  4-in.  pipe  200  li.  lonj;  witli  a  turbine  146  ft.  above 
<i.i-uiii.  Determine  tin-  veloi  itv  of  the  water  in  tlie  pipe  at  whicli  the 
power  obtained  fn.m  the  turbine  will  be  a  maximum.  .Xssuminj;  tlie 
etiiciency  of  the  turbine  to  be  S5  per  cent,  determine  th<;  power,  / 
b  ins;  .005.  .l„s.  19.92S  ft.  per  sec.  :  27.1  107;  II. 1'. 

(»).  A  pipe  12  ins.  in  diameter  and  900  It.  loiij;  is  used  as  an  inverted 
sipliou  to  cioss  a  v.dlev.  Water  is  lead  to  it  and  away  from  it  by  an 
aipieduct  of  rectangular  secti-ui  3  ft.  broad  and  running  full  to  a  depiji 
of  2  ft.  with  .III  int  linatioii  of  1  in  icxx).  What  should  be  the  diflerence 
of  l<-vel  tietweeu  the  end  (.f  one  .Kiuediu  l  and  the  beginning  of  the 
.-tlier./  being  .0064  for  tiie  pipe,  and  .ooS  for  the  aqueduct  ? 

Alls.  14.39. 

70.  Water  II. ws  through  a  pip(-  20  ft.  L.ng  with  a  velocity  of  10  ft. 
per  second.  If  the  flow  is  stopped  in  .'.  .-second  and  if  retardation  during 
the  stoppage  is  uniform,  find  the  ir^.ease  in  the  pressure  prodnceih 
'.i'  =  32  and  the  density  of  the  water   =  62.5  lbs.  per  cu.  ft.) 

.Ins.  62i  cii.  ft.  of  water. 

71.  An  hydraulic  motor  is  driven  by  means  of  an  accumulator  giving 
750  lbs.  per  scpiare  inch.  The  sup[>ly-pi|)e  is  f/xj  ft.  long  and  4  ins.  in 
diameter.  Find  the  ma.ximuni  power  ailainable,  and  velocity  in  pip.. 
(/  =  .0075.)  J„s.    242.4  H.I*.  ;  21.203  ft.  per  sec. 

72.  .X  2-in.  In)se  conveys  2  gallons  of  water  per  second.  Find  the 
longitudinal  tension  in  tlie  hose.  A>is.  9.18  lbs 

73.  Find  tile  pumping  II. I',  t.,  deliver  I  cu.  ft.  of  water  per  .second 
throii-h  a  l-in.  nozzle  at  ^w\  of  ,1  3-111.  huse  200  ft.  Iong,/being  .016. 

Alls.  97..135  H.l'. 

74.  I  he  surface  of  the  water  in  a  tank  is  2S6  ft.  above  datum.  The 
tank  is  connected  by  a  ,4  in.  pi|)e  50)  ft.  long  witli  a  36-in.  c\lini!er 
170  ft.  abiive  d.itiini.  Find  (,n  tin-  vehu  ity  of  flow  in  the  pipe  for  which 
the  avadalile  power  will  be  a  in.i.ximum  ;  1/')  the  power.  If  the  piston 
moves  af  tiie  r.ite  of  1  ft.  per  minute,  find  (<  1  the  pressure  on  the  piston. 
Also  (inil  the  height  to  which  tin-  w.iter  would  rise  it  (,/)  the  cylinder 
end  of  the  pipe  were  open  to  the  atmospheie  and  if  o)  the  pipe  icrmi- 
n.ited  in  a  nozzle  1  in.  in  di,inu!et.  n<-glecting  the  friciional  resistance 
of  the  n.  /zli-.  Finallv.  Iind  i/i  the  power  recpiired  to  hold  the  m  zzle. 
(Coelf.  of  friction  --  .005.)  Ans.i,i)  8. 93  ft.  per  sec;  \/>)  6.85  IIP.; 
(r-l  22.8  tons  per  >.].  ft.  ;  (,/)  3.74  ft.  ;  u)  103.S  ft.  :  (/)  70.8  lbs. 


HXAMPLl-S. 


2\<) 


75.  A  3-111.  li^sc,  4^j<j  11.  Ml  Iciigtli,  tcriiiiniitc';  in  a  iin.  nozzK ; 
W.iter  ciiUTS  the  liose  uniicr  a  head  of  297.^  ft.  Find  liie  velocity  of 
etHux,  tlic  lieiglit  to  which  the  issiitiiK  jet  wiil  risf.  llie  prcssurc-he-nl  at 
liir  ii.z/lc  inlet,  ..ikI  tlie  force  required  to  hold  the  hose,  /  being 
.<x/)25.         Am.  i;8  ft.  pir  sec;  ^Jf)  it.  ;   l8,43rl  li)s.  per  sq.  ft.  ;  gSJ  Ihs. 

76.  A  reducer,  10  ft.  long,  conveys  400  gallons  of  water  per  niin- 
nte,  and  its  diameter  diminishes  from  12  ins.  Us  6  ins.;  lind  tlie  total 
loss  of  head  line  to  friction.  -'«■>•  05529. 

77.  A  reservoir  is  to  be  siip|)lied  with  water  at  llie  rale  <;!  Il.otx." 
-allons  per  minute,  through  a  %'erlical  pipe  30  ft.  high  ;  find  the  tnini- 
miim  diameter  of  pipe  consistent  with  economy.  Cost  of  pipe  per  f(jol 
-^  %ti.  d  being  the  diameter;  cost  of  pumping  ---.  1  cent  per  W.V.  per 
hour;  original  cost  of  engine  per  W.V.  -  Sioo.oo;  add  10  percent  for 
dcprcci.ition.  iingine  works  \z  liouis  per  day  for  300  d.iys  in  tiie  year. 
/■  l)emg  .oo<')4.  ./«.v.  4.375  ft. 

78.  \  city  is  supplied  with  water  by  means  -f  an  aipieduct  of  rect- 
angular .section.  24  ft.  wide,  running  4  ft.  deep,  and  sloping  I  in  2400. 
{)ncji>urth  of  the  supply  is  pumped  into  -.1  reservoir  through  a  pipe  yyo^ 
ft.  long,  rising  25  ft.  in  the  first  1500  ft.,  and  7;  ft.  in  the  second  1500  ft. 
The  pumi<ing  is  effected  by  an  engine  burning  2.\  lbs.  ..f  co.d  per  II. 1'. 
per  hour,  and  working  constantly  through  tlie  year.  .X  percentage  is 
to  be  allowed  for  repairs  and  maintenance;  tlie  cost  of  the  coal  per  ton 
of  2000  lbs.  is  S4;  the  prime  cost  of  the  engine  is  §100  per  11. 1'.;  the 
etficiency  of  the  engine  is  \  ;  the  coel1icient  of  pipe  friction  is  .Qot,\,  the 
cost  of  the  iiiping  is  S30  per  ton.  Determine  the  most  economical 
diameter  of  pipe,  and  the  III',  of  the  engine.  /  being  .0064  for  the  pipe 
and  .oS  for  the  channel.  -'"J-  ^M  ft.;  45^^-455  ^^-l*- 

79.  .X  vessel  with  500  >().  ft.  i>f  surface  experiences  a  resistance  of  150 
lbs.  per  sq.  ft.  when  steaming  at  5  knots.  Mow  mucli  H.I',  will  be 
•djsorbed  in  friclion.d  resistance  by  a  vessel  with  10.000  sq.  ft.  of  surf.ice 
steaming  at  18  knots?  -''""•   ='4oA. 

So.  The  peiformaiiccs  of  two  similarly  designed  ships  are  to  be  con. 
pare.l.     The  one.  with   a   length  of   300  ft.  and  a  displacement  of  Sooo 
.ons.  is  to  steam  at  20  knots.     What  should  be  the  length  and  displ.ue- 
ment  of  the  other,  which   is  to   steam  at  21  knots?     Compare  also  the 
l.H.I'.s.  -'^'"-   3.5"^M'-   '0.720  tons;   1.34. 

81.  From  a  central  junction  four  mains,  each  10,000  ft.  long,  leid  to 
four  reservoirs,  A,  M,  C,  D.  the  waier-leveis  in  A.  H.  C  Ix-ing  r,oo,  400. 
and  200  ft.,  respectively,  above  that  in  D.  If  the  diameter  of  eacii  in.iin 
is  12  ins,,  find  (a)  the  efTective  head  at  the  junction  and  the  velocities  of 
flow.  If  the  velocity  in  each  main  is  5  ft.  per  sec.  find  (/>i  the  effective 
head  at  the  junction  and  the  diameters  of  the  mains. 

Ait^.  (,»>  300  ft  :  8.66  f/s  in  highest  and  lowest  mains;   5  f       n 
intermediate  mains. 
(b)  300  ft.;   4  ins.  for  highest  and  lowist  mains  ;    12  ins. 
fi'r  intermedi.ite  mains. 


■ 


(  II  \n  i  k    in. 


I'l.iiW    I  >\-    WA  I  I  K    !\    ol'i 


^   <  n\\.;i;Ls. 


I.   Channel-llow  Assumptions.-. \   transverse  section   ,,f 
the    uatrr   (Inuin-    ,„    an   np^n   J.ann.  i    n.av   be  suppose!    f, 
o.ns.st  .,f  an   infinite  nunit.cr  Mfd,.,ncntarv  arras   representin-' 
the    sect.onal    areas    of  Hui-I    lllainents   ,„■   stream-lines       Tlie 
velocities  of  tluse   stream-lines  are  very  different  at   .lifferent 
!.".nts  of  tlu-  sanu.  transverse  section,  an.l  the  <listriln,tion  of 
the   pressure   is    also  of  a    e..mplieate.l   character.      C.cnerallv 
siKMk.n;^'.  t!,e   si.Ie   an>l   l,e<l    of  a   channel   e.xert   the    -Tcates't 
retarding  mlluence  ..n  th.-  (low.  an.I  therefore  alon-  these  sur- 
faces .ire  to  he   found  the  stream-line,  of  „unimum   velocity 
Hie    stream-lm.'s    of   maxinuM,,    velocitv    are    those     farthest 
renmve.l   from  ret..r.!in^^  inlluences.      If  the  stream-line  veloci- 
tu's  for  any  unen  secti-.n  .,re  plotted,  a  series  of  ,.,,Mal  velocitv - 
eurves  ,nay  I,.  .■ht,.ined.      In  a  channel  of  svnunetric.d  >e.ti,',„ 


\ 


/ 


Klc. 


124. 


the  .jcpth  of  the   stream-line  of  maximum   velocitv  lK^n^    the 
v.ater-.surface  is  less  than  one  fourth  of  the  depth  <if  the  w.iter 
wlule  the  mean  vclocity-curvc  tuts  tin:  central  verti  ,  |  line  at 


CHMNNhl-hlOlV   ASSIMPIIOS^.  2JI 

;i  ]KiiiU    1k_1m\v  tin-    surfacX'  ali'MH  tliic  c  tKiirtli'^    of  tin-   drptli   m( 
tin-  water. 

In  tin-  iirdmaiy  tluory  nf  IImu  in  (,]hii  t  liannul-,  tlic  varia- 
tKin  (if  velocity  from  jxiinl  ti>  pmnt  in  a  transverse  section  is 
disregarded,  ami  it  is  assumed  that  all  the  stnani- lines  are 
sensibly  parallel  antl  move  normally  to  the  section  with  a 
common  velocity  eijiial  to  the  mean  velocity  of  the  stream. 
\\  !tli  tins  assumption,  it  also  necessarily  follows  that  the  di--- 
tribution  of  jtressure  o\er  the  siclion  is  in  accordance  with  the 
liydrostatic  law. 

Aj^ain,  it  is  assumed  that  the  laws  of  lUiid  friction  .ilready 
enunciate<l  are  applicable  to  the  llow  of  water  in  o|)en  channels. 
'I  luis  tile  resistance  to  flow  is  proportion. d  to  some  fmiction 
<if  the  velocity  (/V:)).  to  the  area  i  .S'  )  of  the  wetted  surface,  is 
independent  of  the  pressure,  and  may  be  expressed  In  the  term 
.S  .  /■,:  I.  An  oli\ious  (iror  in  this  assumption  is  that  ;■  is  the 
iiutiii  velocity  of  the  stream  and  not  the  velocit\- of  the  stream- 
lines alon^;  the  bed  and  sides  of  the  ciiannel.  In  |)racti.e, 
ho\\eve;,  the  errors  in  the  formula'  based  upon  fhesi'  imperle.t 
hypotheses  are  lar^'cl\-  neutrali/ed  bv  f,'ivin';  suitable  values  to 
the  coetTlcient  of  friction     /"  i 

When  a  constant  vi'lume  kJ)  of  w. iter  l<ids.i  ch.mnel  of 
j^'iven  form,  the  watir  assumes  a  definite  dei)th,  .i  perm.iiu  nt 
rej,'inie  is  said  to  be  est.dilisheil  ,ind  the  How  is  sltit,iv.  If  the 
transverse  section.il  .ireii  [A)  is  also  const. int.  then,  sime 
0  ~  :.l,  the  \elocity  ;■  is  ronst.int  fmm  section  to  section  and 
the  flow  is  said  to  Ik  uniform.  l'sii.ill\-  the  section.il  .in  i  J 
is  \arial)le  and  therefore  the  xelocity  ,■  .lUo  varies;  so  th.it  tin- 
motion  is  ste.uly  with  a  varyinj,'  \elocity.  Any  convenient 
short  strett  h  of  a  cli.innel,  free  from  obstructions,  ni.iv  be 
Selected  and  tre.ited,  without  error  of  pr.ictic.il  inipoit.mci  .  ,is 
bemj,'  of  a  uniform  section.il  are.i  e<iii,il  to  th.it  of  tlie  nie.in 
section  lor  the  wliole  lenL;th  imdt  r  i  onsi.'i  r.ition. 

3.  Steady  Flow  in  Channels  of  Constant  Section  \.l  .  — 
The  flow  is  cMdently  unilorni :  .md   since  ./    is  const, mt,   the 


-s//   7/i>     /-/OH-   /,v    (7/./.\\7/.V 


i»l  ■!• 


'l.ntli  Mftlu>.atrr  i.    aUn    .  ■  .n.t.nU.  s, ,  t  l,.,t    thr    u  ,,t,  , -urfac, 

;'  / >  '-'  !'.-r.ill(!  t..  ihr  ill, Mill,  l-h.-.l. 

^_  *  'iii-pli  r      a-     ])cirtiMM      ,,f     tlu> 

i  ■  ^tr.ain,    .,t    Icii-tii    /.    lutwirii 

till-     tu.  I     Iiaiisvcr-M-     SL'Ctiniis 
.1.1,  />/'. 

l-<-t    /    lie    tin-     nulin.Uiiiii 
''^~~~~         "'    tli<'    lu-rl    lor  uatLT-surfiicc) 
ti  1  llif  li( iri/c>n, 

lAt    /'!„•    t'„-     Ictl-th     nf   thr    uctud     IHlillUl.T    ,  ,1"   a    in.ss- 

scction. 

Ili'ii,  Mm.-  til.-  motion  is  imifonii.  tlu  .  st.mal  torcos  act- 
inu  upon  tlu-  nia..  lH-t\vccn  ,>,!  and  M  ,„  tlu-  <li,v.tio„  of 
motion  must  Ix-  ni  <-i|uilil)rinni. 

I  lu-si-  loll  r^  ari- : 

(I)    rih-  icnnioiunt  of  tli<-  uai-l,t  -.t  tli.    ma-s.  \i/., 


I-i.:.  ijr. 


ft: 


•■•-//-itl     /  .-,..///  ;,..//  ._,_//,_ 

//  l)(jin^  tlu-  (ail  olk-vi-l  in  tlu-  Icn'^tli  /. 

N-  'I  I  .     \Vli,-„  ,  i,  sin.ill,  as  i*  usually  tli.-  .  ;isr  in  str.-ams. 

'.  .ippr.  ■xiinati-lv. 

:      I  111-  luv-sur^-,  n|M,n  tin- anas  „„  an<l  />/>,  uliiili  <-\  i.l,-iti  |\- 
nrntiali/f  lacli  otliir 

;i   Tin-   fnVtional    r.sistaiu-.-    .l,-v<lo,„,l    l,y  tli,    si.U-s  and 


Iv.l.  vi/.. 
I  k-IUc 


or 


/'././■..■•). 
■i'Ali       J'/l\r,    :  o, 


w//, 


VI  Noiii;;  tlu-  Ii>tlranlK  mean  depth 


SI  rill)    1 1  Oil-  IS  ■jL-iwiis. 


-'3 


It  11. .u   rein, nil-  {'<  (Ut(  riiiiiir  t!ic  tMi'iii  ct'tlu-  rmuti.ui  /■(:■). 
In  .ir.lin.irv    I-,nL;li>h  iiractiLL-  it  i->  usu.il  to  t.ikc 


/•■(:•, 


/■ 


-K 


f'hv'm''  tlu-  iDctficiriit  nf  frictinn.       I  lirii 


nti. 


I 


/  2g 

V       \  /    J-    (  nil       c  I  mi. 

C  liciiij;  a  coffficicMit  wlmsc  value  (1l]k-ii(U  iipon  the  roii'Miness 
of  tlic  (  ti.mtul  ^nifai  c  ami  u|ii)ii  the  fipiiii  i>\  jt-.  traii--\tTsi' 
si-itinn. 

iliitcit.il  head  //in  a  stream  is  made  up  nf  tvs  <i  |)ait-.  tlu- 
one  Ik  111^,  utili/r<l  ill  priiduun:;  the  \i'l<)cit\  of  llow  and  the 
"tin  r  IjciiiL;  absnthcd  in  ftietional  rcsistaiu  <  I  luis 


It  - 


-A 


;   /-ir, 


Ill  liiTi'.;  I  iMiim  U  and  in  lut  is  in  w  liieli  the  slope  ( i(  th,    l>ed 

docs   not  i;xcf<'(l    ^   It.  per   mile  the   term  is   \crv  small  as 

-.C 

/  /•'.•■, 
eomp.ui  cl  with  .uid    m.iy  he    thsre^arded  without   seiisj- 


l>ic  error.       1  n  this  >  .i ., 


II 


>  /(r') 


/// 


Hx.  I.  A  I  liaiiiii'l  of  rc^iii.ii  ira|H-/(il(lal  si-ciioii,  witli  t)iiiiks  sl<i|iiiiki 
at  30*  to  the  vcrtiral,  lla^  a  bottom  width  of  8  ft.,  nnd  n  wiilih  of  16  fi.  ut 
th<'  fri'f  Hiirlacc.  It  convoys  :8S  eu.  ft.  of  water  per  .>te<'.,  ami  the  fall  \* 
I  ill  3000.  riiiil  the  iiiran  dcplh,  the  mran  volocitv  of  llow,  and  the 
cocffleiciitu  /  ami  < . 


n 


;i 


-\. 


334 


EXAMPLES. 


Uc;:lii  of  w.itorway  =  4  t:m  (o     -44;  ft. 

-■'    =    ,-•**  -^    I'l4f'3    r:    4.Sf'3  S(l.    It. 

/'=8+2X4Vj  sec  30°  =  24  ft. 


TlKufiire  the  mean  dcplli  --  '      —  j^  3  ==  3.4'J4  ft. 

::i  3  :-  3.464  ft.  pt-r  scr, 


Till-  mr.iU    Vlioc  l!V  <if  il'HV    = 


4''<i  i 


I  IcfH  I- 


=..  r,-. 


1 


-■    2000 


Thrrif(]r' 


/  =  .009237     and     <-  =r  82.^13. 


Kx.  2.  HiAv  nui(  li  water  is  lonvcyed  away  hy  a  hnri7:)tital  trem  ii 
10  ft.  wido.  the  dcptli  of  tlic  wntor  at  cniraiuc  bcinK  5  ft.,  and  the  sur- 
faic  falliiij,'  I  ft.  ill  2400  fi-rt.=     (Take  f  •=  .oo!-'.) 

.Area  at  upper  end  -  50  -q.  It.;  at  lower  end  r-  40s,].  ft.; 

the  tiuiin  area   -  _  ("40  +  51)1    -  4;  mi.  (t. 
Therefo.e,  if  (J  en.  ft.  are  ri)riv<-\ei|. 

iliat  at  !)Wir  end  : 


and 


;  vel  mUv  at  upper  end  ;      ' 
5"  11  ^^ 


uifiiii  Vrlocitv 


Q 

45" 


1  he  wetted   perimeter  ==  20  ft.  at  upper  end  ;   =  iS  ft.  .it  low  .-,   .nd  ; 


ini'an  wetteil  pei  linetrr 

Thus  tile  liydraidic  mean  depth  ;//  = 
Hence 


-i>   +    lN>         19  ft. 


45 


(.4 


us'    f'4' 


Q  =  38y  cu.  ft.  per  s<'e, 

3.   Retarding   Efft'ct    ol  Air,  etc — TIil-  nt.inlinj,'  iffcrt  of 
the   air   iiiHiii  tin:   Iric    ■^iirl.n.c  «.t   .1   rivir  cjr  u|   the  water  in   a 
canal  or  in  aiivthaniiel  iias  not  \(  '  been  atiuratcly  <letenniiie(i. 
It  ni.i\'  be  assuineil  that  the  resistance  iier  unit  of  free  surface 


Mi. 


h'l  /./AV'/.Vf,    I  IIF'Cr   ()/■'   .//A',  HIC. 


225 


wilti-il  pcniiu't'  i   l)cciinu'->  /' 


-liiL-  tn  the  ,ur  i^  al)iiut  f''/<'  A'////  c  if  the  rr-'ist.uKc  cluf  to  similar 
units  at  tile  i)i'tt"in  and  >ulc--  nf  ^iniinth  (.lianiU'U.  I'luis  if 
.V   is    till     wicitli    lit'  tlir    tri_<'    surtaii.    in  a   simxith  thainul.   llic 

.\ 
lo' 
In  i;riU'ial,   tin-  Nscttt'il   priinirUv  nia>'  1>'     cxprc-^scd    in  tlic 

form  /'  -^        ,  l>  liiin"    K>  tm  --iniKiili  clianmU  and  l;h  ati  r  tiian 

/>'  ■ 

If)  fnr  I'lir^h  LlianiuU       'I'li'j  v.ilin-  n(  /i  \-.  c\i<lcntl\-  diminisiird 
li\-  (ipnif-iiii^  wind^  and  increased  liy  IdIIiiw  in;,;  'Aiiuls. 
A;_;ain,   itl  tlir  tonmila 

/■>  r) 
nil  ~ 


,■/ 


w 


.1   / 


'\ 


arc    --nnilarU-    n-l.itcd    in    tin-    drtci 


V   /'I  V  '  /.' 

minatioti  <4  r.  tin-  mean  \cl<>i.it)-  of  How.  If."'  i^  (.-on^taiil ,  the 
|iiodiKt  ///.'  mii-t  aUo  l)r  i.in-.tant,  >o  that  if  w  iiurcMsi-s  /  nnisi 
dimmidi,  and  ruf  ;iis,i.  \'\m>  in  .1  very  llat  >oniit!\-  tin- 
llow  ini\  Ik  m.iitii  Hill  d  l)\  makinj;  /;/ NiifVii.  ient  Iv  iar,L;i-,  while, 
aL;aiii,   it  the  i  haniiil   ln-d  i-~  ^tff|>  ,."  is  small. 

1  hi     iio-,iiiii    i.ai:-.,d    li\     a    w  attTCourso    ilKTCitSi.  -    witli    tilt 
.•■■ipiditN  of  il.iw        At    th.    ^aiin   tim.    the  sectional  area  (./)  of 
tllc   \vatirua\     al-o    iiu  reases,    so   tlia'    tlic    vtjoiitv    of  llow    ;• 
(liminislu-^  1  lin>   tlu-re   is   ,1   tetukiiLV    to   ap|iro\iniatf    to   a 

••  |)(rniani  nt  n  -iine  "  uiiiii  the  renistaiKe  to  erosion  iialaiues 
till'  tenihiK)'  t<i  siDin. 

ll.iiK'.  tliroiii^hont  any  lony;  streli.  li  ol  a  n\tr  passinjj 
lliroui;li  a  specilk  -^oil,  tlie  mean  velocity  of  (low  will  he  very 
nearly  constant  if  tlie  amount  of  flow  {(J)  does  not  vary, 
(ienerally  spe.ikm-.  the  voluiiu  i.inveycil  li\'  a  river  increases 
from  source  to  mouth  on  account  of  the  ailditions  received  from 
tnhut.iries,  eti .  ."^ince  (.'increases.  .1  must  also  increase;  and 
if;///orr  is  t<»  rem.iin  constant.  /  must  ihmmish  It  1  ■  to  i)e 
observed  thai  the  surface  s|ope>  of  iar^;e  rivers  dimmish 
L'radnaHN-  liom  source  to  moulli 


fr?-  ^ 


1- 


-26 


Rl.r.lKDIS';    ill  EL  I    (V-    .-UK.  I.IC. 


•o\  .1   -i\rii  .lisJi.ir-.-  (Cj)  tiu-    iiiraii   .Irjitli  (/,•/>  (limiiii-;li 


.IS     /     IIUMM- 


til 


c   i.()>t    111    inn-.triu  tiiiL 


il 


ipprM\iiii,it(_i\-  i,i..].,irtM'nal    t..  the    iikmii  .kptli.  it  i-  .ul\i-ai)lc 
>il  .is  lar,:,c  a  >1w|k-  as   pnssiblc.       Hut  the  \i  l,K|t\• 


'  i  \  c  I  i 


11. 


iiu  itasi-s  w  itli  i.  an 


tiu 


^i. 


jir    must    tliLii'ti  lie 


not  cxci-i-d  that  Inr  w  hn  h 


(1  1) 


'rcat  as  t, 


1  cause  tiK 


'■'Ksidii    .^1    the    hanks.      On    the  ntlicr  hand. 


must    n.it    h 


ill 


as    til    allow 


the 


ik  [)iisitii  m 


.f 


it!)    nt   .Kjuatie    plants    cir 


.1"  til 


mil 


.111(1   (itluT   detntii' 


\\  llU  11     W  nllkl 


(lull    nllstlMkt    til 


e  w  aterw  a\-  .iiii 


IIKI 


>t  III  iiiainteiiaiK'c. 


I'.et\ 


I  ail.l 
tl 


)e    \.iiieil    in   .iii\-  III  111 


I  iiillMiUr.ihle  Item  t'l  tile 
ween  these    evtienu-  limits  the  sl.ipe 

nu- 


lled 


III. inner,  the   iDiitnill 


ences  heiii''  the   eont' 


'i^'uratii.n  '^f  tile  ;4ri)iind  and  tlie 


nv^  111 
nature  ut 


tilt 


.i!   tl 


irtiii''M   which   tlK 


passes. 


.ireful  (ieterniiii.itiiiii    shniild  lu-  made  of  ti 


it  tile  tiirii-  elements 


.mil 


.; 


w  liieli  wiiuld 


disci 


l.tr; 


In    I 


r.iiui'  thi- 


ll 


Irmii    I.]  til  JO  in   io.(X)0.  and  l!ie   in, 
iii.i\-  be   cmisider.ihle  wlieii   liie  c.iii.il 


lis     |),i\e 
e'liitude- 


In    i\ery    case    ,i 
le  best  I  iimbinali.iii 


;i\e  a  speciticd 
■  lopes     v.irX'JM:; 


liotll   V  .111 


throiHfli    M 


t!ir<>iii;h  .i  \\ell- 

ActunlillL;    to    HeL'iMiid    the  \,diie  i 


ip.icted  m.iterial  c.ipable  of  resisti 


particli 
sh 


>t   lo.im   should  excel 


eriisi-ui. 
it    r-  for  u.iter  c.irryin-  tine 
I    It         _\;   111.      per  second,   .Mill 


slloiild  not  be  le 


-s  th 


111 


are 


men    u  1 


til 


co.ir.se 


J  It.     . 
particl 


per   sec< 


if  tile  waters 


if   1 


water,  tile  ;^'ro\\tli  of  weeds,  etc.,  uh 
teri'  witli    the   llow ,  is    pieMiiti 


o.iiii   or   s.iiul 
ii<l 


In   cl 


c-.ir 


IK  11  Would  serious|\-  inter- 


il    the    \elocit\ 


if  n 


ow  Is  limn 


to 


It. 


111.    to    .,S    111. 


11 


J)er  siiond 


le 


sioi'e   o 


(ii/ii 


liih  /, 


m    Willi 


troiil 


)le 


t.)    I 


)e 


anticipated  from  i)lant-^nowth,  iii.iy  b 
am!  iiia>-  even  f.ill  to  i    in   io,t)on  when   tl 


as  sm.iil  as  ^  m  lo.ooo. 


le  w. Iters   .ire  e\i  i  p- 


tion.ills  cle.ir.  .IS  ill  ilie  case  of  the  .uiuediicts  on  the  iJluiis  .mil 
V.uiiie.  <  >ii  the  other  hand,  the  slope  sliould  rarely,  ii  ever, 
t'xceeil  \2  in  lo.cxio,  ami  .is  a  j;ener.il  rule  the  slope  should  be 


li 


CSS  th.m  U)  in    io.dck). 


Me   ordin.irx   t  Ii.mnel  toiiiii 


V  —  1   \  mi,  is  .ipplicahle  to  the  (low  in  .i 


"I  I /nit. 


1);  .is  till 


RIlTAKDlMj    lit-lEf.r   Oh   AIR.  I  TC. 


coiuliiit  il'if<;  iKit  rini  lull,  ami  ^inci'  :  i^  [iinp,  ntiun.ii  ti>  \  ./;  it 
i-  a  niaximuin  lor  sumo  dilMiiU;  liipth  ■  it  watir.  When  iIk- 
\\,iti.r  tills  the  conduit,  llu-  lormula  t'<ii  v  h.mnei-tli 'W  mu-Iu  to 
clian,L;e  -luiilcni)-  s, ,  a-  tn  ai^rn.-  with  th.it  t'lu  pija-llow  .  ami  in 
this  respect  the  tlu(ir\  is  tluiiloir  iuipcrti'tt  liu'  intMii 
VL-lncity    ot    tli.w    111    a    lomhat    -liquid  not    Ijc    11----   than    ahout 


ft,     iO. 


iiir    second,    .u\i\    iiia\     lie   as 


ifi-at    as 


^4 


It. 


(1.5    in.  i    per  scci 


liL;h    \iloiitiis    .naljlc    the  waters    to 


<  arr\'  oti' floatiiiL:  delin-  and  ^aml  particles.  'J'liere  slioulil  he 
no  sudden  ch.mL^es  of  slupe  or  of  section,  as  tlie\-  fa\'or  the 
torination  of  eddies  and  the  ileposition  of  detritus. 

i  he  follo\uii|4  t.dile  of  -,lope^  .ind  nu  an  \elocitics  i-,  taker, 
from  tne  article  hy  i  ),un  in  tlie  iiiivcl.  Sc.  lies  Aiik'- 
Mciucjirc; 


Crapunne  C;inal        . 

M.ir.scillc 

Carpi  ntras     "      .... 

Sainl-Marliiry  Can.il 

VcrilDii  Caiial 

Neste         ■■      


Hcaucaire  Canal.  ... 

l.ariM  he  "      

I'Oiircq  "      

Uhuiii  AqueUuci  .... 


SU)|<«*  in 


lu 

3  I"  7 
J  I..  4 

5-4 

1  ?  I"  2 

} 

2  lu   I 
2 
1.2 


Mr.tn  \  t  Uii  lly  per 
Sccoiul. 


Avri-  Aqueducl. 


Naples  Aqueduct  . . . . 


Moiilpellier  Aqueduct 
Croton  Aqueduct 


Ic.l 


Metres. 


3-3       j         I 

2.;)  In  f.tr       .77  lu   2 


as 

3.1 


1.(.4 

J  (.4 

1.21 
1.07 
1. 31 
1. 18 


.'■3 

3.62 
2,36 


•  5 
•76 

•5 

•37 
■  51 
■4 
■3'- 


.8 
•  7a 


Salurc  «if  i  anal  Sjdc^. 


Mhivial  soil 
r..iri!i  anil  ruck 
\'i-f;flablc      Soil,     fis 

surrd  llinistune 
Clay 

Calcareous  rucks 
Cl.iv.   puddinK-sionc 

rock 
Alluv  lal  siiil 
CalrareiiU"!  rocks 
Karili 
.MillMone-Kril  mason  - 

ry    with    a    jj-inch 

I—   .02    ni.)    facing 

in  cenicni 
.Milistone-^rit  muiion* 

ry    with    a    jl'i-inch 

(=  .03  in.)  facing  in 

cement 
Kulihle  masonry  willi 

a  ^  in.  (=  .015  m.) 

facing  in  cement 

Rubble  masonry  with 
a  il  in.  {rz  .oij  m.) 
facing  in  cement 


I 


V    t    : 


til 


lilt 


-^  ').V    THi    /YWU   OF    IHF.    SFi.TlOX   OF    7    0/7VA/:A. 

4  On  th3  Form  of  the  Section  of  a  Channel.  -  j  l,.  ,u„<i.- 
mcnul  U,rmui..  ,,.v.rn,n^  tin-  ,;„„,  „r  the  transverse  scai.m 
"t  .1  chaiiiiol  arc 

(J  =  ./.- 


..     .         ,  J  . 
1  iiercto' •••.  alsn, 

l-"-T  ehaimols  nt  the  .w?w/,-  .v/,./.,- 


Take  r-  _  u»i.  ,?  beini;  some  C(>n-.taiit. 

Then    if./;,  is  a  Mnall  chan.a-  n,  tl,e  velocity  corresponding 
to  a  ,n,a!l  chan^^e  ,/,..•  ,„  the  Imiraulic  mean  depth. 

2r  .  ,/.-  =  ,/  _  ,/,„_ 
ami  therefore 

>/'•  _  i/iii 
:■    ~  2111' 

Thus  the    hydraulic   mean   .Irpth    nnwt   !,.   J,an"e,l  ^,,,,,r 
cent  to  produce  a  chan.^e  of  ,0  per  „,n  ,„  th,-  vel,;,t^•  '     ' 
A^Min. 

r.oi!nvtr  """"p   ""'    -•  •""'  ^'"'"'^'^^  ^  "— -  --e 
apu   V  than   .•  .      hor  exan.plo.  an  increase-  in   the  velocitv-  of 

j-ss    han   n  I>cr  cent  will  cause  an  increase  of  ,0  per  cent  ,n 
the  'lischar<je 

i-or  channel.  -ivi„;r  the  .v,„;.v  ,/,s,/n,ro- 

r        r  ,, 

Imt    a    .::uen   volume  -„   uater    there   must    be    a    sensible 
Chang-   n,   the  slope  to   produce  an  appreciable  chan<,e  in  the 


(l.V    THr.    l^OKM    Oh    THl     s/.V.T/O.V    ( '/ 


CH  -iwri 


i  i  i-  'I 


;li  icitv    of   ll'iw  ,   .ilth()UL;ll 


1,  ,i;ciit,-rall_\'  <i)c.ikiii,L;.  tlic  wtttcil   [i 


riiiu;ti,r    /'i   liiiiiinislK  s   or    iiurca^c-^   .is   /   jikt 


r.l^r-.    ipt    ilillim- 


i>nc--,  .iiui   tiuis 


.mil   tlicrctdii 


./ 


iiu  riM--is    i.r    ihniKii^lK- 


mori;  r.i[iiill>'  tlKin  /.  An  iiKrc.i--c  nl  n,  jur  •  nt  in  Ilic  \(.-- 
locitv  causes  a  iliiiiiinitidii  ct  ahmit  4  prr  i  ciit  in  tlu  scctiniial 
area  of  the  \v,itL'r\\  ay. 

I'l'i    cIkuiiilIs   of  the   .\ '/;;/(■  .V /('/><    ami   Ljr.inu;    tin-  .uin.i   iii.\- 


i/iarc.    J' 


aiiil    ai--o     , ,     are  i  on 


/■ 


-taut.       A  tnrti 


It  r  1  c  nilith  11  1- 


ni|uircil  hrtorc  the  scitimial  area  lan  he  lii-tcrniiin  il 


ris(il;i  \\i   1.       A  i.anal  ot    rectaii^'ular    scetn 
X    is    to    eon\'.-_\'    \\,itfr    ot    ilrptli    y 
\\\\i<n    that    ,  //'/.  ;■   the 


iiiii  of  width 


uith    the   eon( 

set  til  in, il  ana  1./  1  of  the  \vatirua_\' 

is  lo  lie    ,1    I  oil'-tant   iiii.intit'.    rr  the 


wetted    iierini 


nuniiniiin. 


■tel        /■ 


1^    to    lie    a 


It  )■-   ])io|H.^(il    to   tmtl 
the     Illation     lietweeii     1     andi     1     so 


Fin.   12^. 


that     .:!) 


the     \(lcnit\     .if    tl 


I'W    ina\-   he   .1    niaxinniin.    i/i   the 


i|iiaiUit)    111  ilow   nia)'  he  a  maxinuiin. 


i'w 


X'r 


.    t 


Q^  A' 


I  A 


If 


IS  a  inaxinuiiii, 


Ifc 


IS  a  I1la\innmi 


(• .)  -"  =  '"- 


r  .,i.\  ~  A  .  ,,/' 


3J».  r.Li  -./'.  ,iP 


F" 


I2II 


;!f    t 


II' 


\ 


-\?o         ",V    />//:•   iHRM   ()/■•    THi:   >7;<;7/(),V   O/-    ,^    CH.-tSSFL. 

In  each  .asc,   if  ,^  /  ^  o,   i.e..  if  the  area  i>  iu:wtant.   tlietl 

^/Z*  =  o : 

and  if,//'        o.  i.e..  if  the  netted  penineter  's  a  nu.uiiunu.  then 

,f.  I   n=   O. 

'I'hii^    tile    .ame    results    are  ulnamed  f,,r  tlie    ;,rM(,leni  m  its 
different  ^onditifMis. 


./ 


Tlierefor' 


jnd 


I  lein  L' 


-rr     and      /'  -  x -J^  2}\ 

■iP  =  //.l-j-  2  .  c/j'  =-  o, 
X 


Tlierefore.  al^o, 


rn  =; 


»- 
r- 


««. 


,ind 


/'-   2' 

/.vl 


2   »  2 


A  .uital.le  \ahu-  f,  .r  ,  o  .rres,„„uli„t;-  to  the  slope/  or  to  the 
^'■''"''  "'  '"  1,=  \]  ^'i"  fx'  "hf.,iMe,|  f,,.,„  !h,.  Tables  of  Ha/ifi, 
Kiitt.-r,  ..r  Maiinin-  .it  the  end  ,,f  the  Ji.iptef. 


riN   THF  lORM  OF   THE  SF(:TlO\   Ol     1   Cll.-IWr.i.        2', I 

I'lsiiiii.iM    II.       'I'lu-    MLticii    i^    Usually   in    the    toiin    nt    a 
(juadnlatctal,    the    uou-parallcl   skKs   sloping;   at    au    .in;4lc,    «, 


feifc,\  ,«»co(^ X 


y' . 


WW 


I.  T * 

Fic.   127. 

(Icpetulin^    upon    the    nature    of  the    'i.  >i!    thnuii^'h    which    the 
channel  passe-;. 

I'or  c\ani[)!e,  in  a  i.anal 

^  =  63^  36', 


'^  =  45°. 


with  retaining'  walls 

with  stiff  earthen  --iiles,  faced. 

with  stitV  earthen  sales,  vuifaced,    0  —  33    41', 

with  sides  in  liylit  or  sandy  soils  ^  :=  26^  34'. 

In  such  a  channel  let  .1  he  tlie  bottom  width  and  y  the 
ilepth  of  the  water.  Then,  the  remaining  conditions  bein^ 
the  same  as  tliose  in  Troblem  I,  it  again  follows  that 


dA  -  o     iind     ciP 


Ikit 


and 


A  —  }<  I-  +-.)'  cot  f^)     and      /'=  .r  +  2r  cosec  ^. 
J'lrst.      If  t*  IS  gi^i'n, 

,/^  =  O  =  .)' .  <h  4-  ix  4-  2y  cot  «)</'. 

(iP  =  o  =  i/.r  4"  2  cosec  f  .  d}'. 

Theretore 

2/  cosec  "  =  jr  -|-  -y  i^"*  ^1 


f 

\^. 

t 

:     r 

^ 

i 

r-  — 

^^^^1 

'^'^H^l 

yj 

■1 

232  ON    THl:    hORM    , 

or 


'A    THF   Slf.TKW   OF    V    r7/,4,\' 


Wfi. 


or 


,in    » 


tan 


:?.Hi     -  CCS  ff), 


2y 


129. 


Th<-    scTti,,,,    may    hv    rasily  sketched,   as  in    Fi-s.    12S 


and 


-■S- 


.1 


Fii;.  i:S. 


>-A„\ 


Li 


I'roni    tile   nndillc    iinint  ('    ,t"    / /;    ti,  .    1,   f*  -i, 

I  '      ■!-;/..  the    bnttoin  uidtli,  draw 

./•atn,d,ta,,,k.stnj/;andc,,ual    ,n    length  t.>   the  depth  ,., 
the  wai    .  .        1  hei. 


An  H 


^^  beint;  tile  1,'iven  slope  (if  the  si'des. 

,.     ^^''''    ^'■'"    ^"■""■^'    ■""'     /<■    -    radiu,    describe    a    circle 
lT.,mthep,„nts./an,l   A'diau   tangents  to  touch  th,s  crcle    .t 
.;and    /:         /../  evi.lently  h.seUs  the  an^le   CAD.       Theref^.re 

tan  '-^':.  tan  r.;/.       '' ''  ^^ 

-  AC    x^/r  ^•■^■ 


Hence    7T  —  (  A/) 
rc(iuired. 
AL;ain, 


■'"')    A/),    /;/■     1,.,,.,,     „,^,    ^,^^  ^^. 


pc 


r  —  CO'-  fi 


^J    =    I'l  2 

,2   —  COS  h 


=  r 


sm  # 


'•) 


i)N    THi;   yOKM    ()/     THI.    SUCTION   Oh  A  CHASNEl .  233 


«?■■ 


y 


_     /  A  sin  H 


ami 


I   —  cos  H 

r  =  2V       -  .         -      +  2)'COSCC  0 

sin  f' 


V 


Therefore 


-  cos  ti  __  2.-1 
sin  0  y 

A        V 


m 


r 


f  2 


and 


the  cneflficicnt  r  bcinc^  oht, lined  from  the  tables. 

ilu-  foIlowinLj  Table  l,mvcs  the  best  relative  \a!ues,  />,■;-  n/iU 
r/  (i/td,  (ft  .f,  J',  w,  and  /',  correspondiiij^f  to  spccitieii  \-.ducs 
of  '',  and  tlie  .ictual  values  ni.iy  be  obtaii.ed  b>-  niultiplxiiiL,' 
tliose  of  the    Table  bv  \  A  : 


« 

.r 

V 

Wl 

/■ 

90' 

1-414 

.707 

3535 

2.32.S 

6o* 

.S77 

.7f)u 

380 

2.fi33 

45- 

.613 

.740 

370 

2.7o«, 

40° 

•525 

.722 

361 

2.77-' 

30°  52' 

■471 

.7"7 

3535 

l.S2n 

35° 

43') 

.607 

34S5 

2.S70 

30° 

.3J<> 

.664 

332 

3.012 

26    34' 

.301 

.636 

,.t      :,lv.;nc     I-.,- 

31S 

3'4-t 
\      ■,,\ni-,t 

actual  practice.       The   char.icte'r  of  the  --'iil.  thi-   importance  <'f 
pri'ventiny  excessive  (iltration,  and  the   I'lttij  ulties  of  construe- 


* 


i 


l: 


-it        (i.v  7///:  /(iKM  ')/    ////;  .s/y;//().v  (-)/•  ,•/  <:ii.'I\nf:i . 

liiiii    ami  ni.imti'ii.iiKc,  ni'.i  ii   iciiilir  it  inci-,^,ii-\-  to  insure  that 
tlir  ilipth  iirtlu-  w.itci"    -li.ill  tint   cvcccd  ,1   n  it.im    limit,  say  .S 
to  i:!  It.  [2   m.  to    ^    III.    .       Ill  l-i.nuc  the  (k-])tii   lit   inij^ation- 
lan.ils  is  ht'twfcn  4  and  (t\  it.      I   2  ni    and  2  in     . 
SicdiuL      If  tlic  iio(toiii  -.culth    I   IS  fix,,!,  tlu'ii 


dA  =  n  "  (x  -\-  2\  Lot  ")t/r 


,/'V 


dP  —  o  ^  2  coscc  ^.  ,/r  —  Jr    .    ,  „    7" 
••  sin-  w 


IIciii,'!. 


or 


ux 


X  -j-  2y  Kit   "  I 

2  COSCC  V  2  COS   W 

X.  sin    **  cos    1^  —   )  '2  COS-  'V  _   I  ), 


and  tlicrc'loro 


-^m  2*^ 

.t  :^     —    V  COS   2", 


tan  (t  —  2'M  tan  2*^ 


If     I: 


It  may  lir  ol)..<r\rd  tli.it  •i'-  'lu'  width  1  r"!  of  thr  l>ott(Kll 
increases.  "  al-o  men  .isc 

If  tlic  widtli  IS  ;///,  tlun  tan  2><  -  x  .-nd  ^  x.-  45',  sO  t'hat 
the  tri.iii'_;ul.ir  section  ol  nuninnmi  |i<  1  nia't-jr  is  a  scnii-scjuarc. 

Third       l>  (lit-  <i,p//i\   :.s  /n,d.  then 


and 


Tljcrcforc 


or 


d.l       I'       i(/i        j'  cosec'  ^  .  </# 
._  cos  ^ 

dP=Or^dX   -    2\      .     .,^-d>' 

■   Sin'  " 

y 

'  .'     (OS     "' 

COS  "    ^  i     and      "   _  6o'. 


O.V     ////     li)KM    (1/      liih    .S/f./70\    ()/     ../    (:H,tSM:l_. 


I 


rkMl;i  i,\i    III.      To    tni<l    tile    proper    ^i  ctiona!    form    of  ;i 

Ch;illIK-l     ot     Ix.ttolll    \M(!tll    2:1    ^o   that    tllr     lllt.m    VcloClt\-  of  tlfjw 

may  hi    i.d!i--t;iiU  tor  ,ill  il.pili'^  of  \k, iter. 

I. it  . I',  J',   l-'ii;.    I  V  I,   111     thr   t  o-oiuiiKit'--,  of  an\-  ponu  /'   m 
•ill    profile  rcfiTri-(i    to  tin     luidd'r  pont  ('  of  .//.'.  tlu-   liottoni 
.i.ltli,  .i-' oriL;!!!.  .111(1  lit   V  lir  till    Kn-lh  ul'.//'. 


V\',.  11". 

."^iiKi'   .-■  is   to  1)1    roiivi.int.   ;/.•  must     iKo   hi     lonst.iiit,    .iiul 
thcrefon; 


; 


/,.,/, 


,1    lOH'^t.     =    111, 


wllii  h  111.1}    hi    u  nlti  11 

I   I    t/.t    --  tttiS  -{-  ,/). 

i   'il  H    II     I  I  O.l  I  I  I  u, 

I    .  i/t    —  ;//  .  rA  ii/ii/.t-    ;    i(y^)<i, 

.iii'i  ill.  nf.  iri 

dx  dy 


'"     "  ,  r        ;/ri* 


Iiitrjjr.itin^j, 


l"t,%  <.''  -f  ♦.>'"'  —  '"')  +  '■ 


I  hi  inj;  ,1  imisf      ,  <if"  intiL;r.itiiin 


Hut  >        .1      ,  Lii  .1  --.  o,  and  ,    .  o  =  \o^^Jll  -{■  ^ a' —  tti*) -\- c. 


Htncf  -  Iol; 


I  liiirforc  _;■  -f-  %  I'-  -    m'  ~  f>c^. 


I 


\ 


S3'J  (>\    Til/:    /(lAV/    ( ','     ////     s/illOX    (  ,■     .1    r/M.V.VfA. 

i  It  IKc,    too, 

Aililin^-  tof^LtlK  I  the  1,1-t  two  (•([nation-^, 


iir 


2y  =  /"(  "■  4- 


)ir 


\  I-' 


or 


y 


I 


wlmli  is  the  (•(]iKitii>n  t..  lln-  k  iinircd  iniifilc,  ati.l  i^  a  curve 
wliiih  Ih1mm-s  tM  111,  Jass  ,,r  latcnarics  ami  ulmli  r\i(!ciuly 
flattens  (lilt  \rr\    r,i]ii(li\'. 

It  thr  lidttom   w  iiltli   1-  su,  1,  that 


r/  ///  l> , 


the  ciiiiatiuti  ')(  I  I  iiiii  s 


:;(•■ 


and  tin;  iinit.li  1  a  tin.  latciiaix  ot  paiani.tir  v/,  with  its  avis 
coincident  uitli  tin  l».tt..iM  and  its  dimtiix  louHiduit  uitli 
tin-   \i  'tn  al   at   tin     iiiidilli     ( .1  tin     s<  ..  tii.i!. 


r  t 


Fk;.  131. 

ruoiil.lM  I\'.  A  I  liamiel  nf  j^i\en  sl(.|)(  has  a  j^iven  sur- 
face width  AC,  vertical  sides  .;/•*(=  r^j  and  i  li  >  —  j)  <>t 
Kiven  depths,  and  a  .utv.l  1..  d  /.'/>(=  /,)  of  given  length. 


0,V    THH.    lORM   or    ////     ^l-:iU)S! 


('/•  .1  <:Htss'i:i. 


I 


'"'"■  -"ii"'!!!!  ,111, i  vrl,Hit\-  ..r  il,,u  M,  til,  ,h,,nni'l  uill  Ih'  ,1 
nKiximuni  ulun  lli.'  tmni  .,t  tlu  l,r,l  /.7>  ,,  ,,  cirvnl,!,  ..r.-. 
I  Ins  cm  1)1    easily  ]ir()\i(|  ,i^  tcil|.>u-, 


Sincr  the  slope   i.   l  n!i-,t.lllt ,    .'      .    i  ;;.' 


x/; 


Hut  /'  I  -:  /,  -f  J'l  +  V.)  1-  .1  ^  "11, t, lilt  'lILUltltN  ,  .111,1  til,  I,- 
f'MC    ,-    ,111,1    ,lls,,    (]   uill    1„.    ,,    iM.lMIUnill    U  lu-ll    .  /    1,    ,,    111,1  Mllllllll, 

HeJHH',   t,„,,    th,    ,iir,i   iKtue.-ll   ill,-  ch.,r,;    /-V  .,11,1  tile  ,  n!\<- 

rmist  I.,-  ,1111.1X1,11,1,,!,  .ii„ltluTrt,,iv  die  .u,a,-  mn,t   l.e  .,  mi.  iil,,,- 

•'"  '''"     1""""'    "'    'li'^    i>,N-    the    C.kiil,,-,    ,,r    \  ,„i,,t„„,,    ,,    ,,s 

(mII.iw-, 


'I", ike   (  >  ,,,   t W 


i'l'ihucl  ,1-,  the    .pi,,;,!,,    (  '(  '   .,,   the    .ixis    ,,1' 


-».   .iii'l  the  \,  ,ti,  .il  thion-h   I  >  ,1,  the  ,i\i,  ,  ,f  j.       The,, 
'■'  /       '■/''   !-■  t,'  lie  .1  m.iMiuum. 


K   ,1   vi\ 


"  'I'MiitiiN  .in  I  >,,,!;;  -_:  _,.^_  iK'  -  1..  ,ii,,i  V  -  f. 


1,1    /       .  y  ^.  ,t  \  \     j     /,'.,/  heiiiL;  -,..,11,    11)11, t. ml 
Ih.n 


./.-' 


.m,l   th,  iel,,it 


th  ,t   I,. 


.iiul  thu> 


'/ '    ,,  t,i  lie  ,1  ,iei  \imiii,. 


,1/^ 


^+''*'  f  ^-^     s^-^r^-^'' 


" '  '■  n 


^  + 


♦  i-f /»^ 


--  e. 


ril 


i   5  ti 

I  •  i; 


O.V    THi.    tORM   Ol-    THI    SH.IH)\    Oh    -l    CH.-IWEL. 


Illcl\-t( 


IntCL^ratiiu 


./x 


<■■  -  r 


dy       p       iV?-(r^-,f' 


X  +  «,  =  »  <r  -  If,  -  yf. 


tlu-  i'(|u,itiim  \"  A  K  n\lr  <it'  railnis  ,r 

Ilciicc  ilic  pMilili    /.'/'  I-  a  I  ire  111. ir  .iic. 

Tlu-  itiaxmiimi  .K  ptli  (.Ithr  cli.itnu  1  j^,  ,    —  ,,. 

I  In-  cniistaiits  ,|,  ,, ,  ,/  laii  hi'  IdiiiKl  from   the  three  concii 
tidii-  that  the   arc   is   of  <;i\Tn  h'n.L,'th  and  lia-  tr.    pa-^s  tliron'^h 
the  l\\  M  fiv,  ,1  |).  .iiit>  /.'  ,in<l  /'. 

'"'-"Ill  ^'    \  ■       Ihi  SiiiiHiniilar  C/-^///^/,/.— I  hcorcticallj-. 

the  best  f(irni  of  ihaiinil  for  a  ^^wv\\ 

"  v\at<ruay  is  oiu-  in  uliith  the  bed  is 

\  ;         a    lirmlar    are      I'ld,.    IV),   as  tlic 

Vx..  y  u.-ttr,l  perimeter  i-  tlu  n  a  niiiiiinum 

.111(1    thr   mean   .K  jitli     or   radin^.   a 

Vl<:.  132.  ' 

1 11  a  \  in  I i : ! 1 1 , 

In  the    .emieire.ilar   ehann.l.    \-v^.    j^j.  j,  t  the   free  Mirfaee 
subtend  an  aii^K    "  at  the  >  «  ntn 
'Ihen 


r    1^ 


ill  ^, 


->-sin^.=:':.(,--/j 


and 


/» =  rfi. 


t  bfiiiK  tli(    raihus. 
Therefore 


m 


A 


in  (f\ 


A       ri        sin  ff\ 


I 


II 


OA'    ////:    lUKM    i)i-     ih/:    .Shi.  IK  i\    i)f-    .1    LUMNhh.L. 


239 


flH'l  ,    villi  (■ 


mi  —  />;■'     -  h 


A' 


i--'«-h 


sin  «\-' 
r-H'\  1   -      „    ]  i  -     U0-. 


If  tin    (iianiicl   iim-,  (till.    "         ,t.   and  tllcll 

r^n-i  =  )MiO'. 

Av  .1  tir^t  .iMpiMviiiialnui  it  nia_\   bf  a-Mimr.i  that 

lipi   -III, ill  ciianiirl  -,(rtiniis  with  CLnicnt  faci^..  /;  —  .00032 

I'h.mm  !    ..|  nuaii  diiiu  u^inn-,  with  viiKinth  tatrs  /'  ==  .00017 
tliamu'Ucpt   larjM-  iliiiiiaisidn-.  I,-—  .oooi  I 

In  MUtiK  nu.i-iin-  these  cm-rrii.  imtv  hi-ii.nu-  .tHjo4,  .o(A)^. 
ami    0002,  rcsiK-ctively. 

Miscillaiiti'iis  J'n'hhiiis. —  I'lu'  Ind  of  tin-  ainudm  t  ,it 
N,iI)1l-s  is  si-ini-rlliptii-,  hut  beds  in  the  tnini  ,,|  .1  -^rnn-illipsi-, 
.1  c>-cloid,  a  iiiiali.il.i,  ,,1  .III  li\iKrl)Mla,  w.Mild  ..iil\  l,,-  adapted 
iindfr  MT)-  <.'.\i-t.'pti.inal  i  ..ndiliMii-,,  .1,  ulnn  a  iiir\cd  pi,,til,-  is 
r.'(|iiitcd  with  a  limited  depth  Ilu  waterway  ;ind  ih.  \\,tt.d 
])erinieter  lan,  Mfeourv,-,  |„-  appr.  ,\  imatel\- calrulat.  d  \vu\v  the 
!.il(U\  n  prupertiev  t,|  the-i    i.  ii.-\es. 

For  the  s,},:i-,  lliMh  urtimi.  if"  ,/  and  h  are  the  senii-mai'T 
and  mtnnr  axes, 


and 


f  where         1 
*  when 


A  ^=  n     , 


I   .iiJS,       1.11,       i.KJif)         1     ,404.        i..A,i.        i,.,n,        I    t,r..        I 


For  the  iyclouial  sution,  if/-  is  the  radius  i.f  the  ycneratiiii; 
tirclc, 

A  ---  inr*     and      /'        S/-. 


I': 


M 


I 


'  ft 

ill 


!i 


240 


1    llLlXtuIC 


y4  QUE  DUCTS. 


»/  ~  'inr, 


■  mil  llu    111  )\\   (.  (|ii.itiiiii  Ixi  I  iiius 

If  tin-  \\,ttfr-liiir  is  at  .  / .  / ,  ditnicil  liy  tlir  ,ini;K-  "  w  !u\  li 
tin  ladiiib  (I. I  <it  tlK-  ,L;riuiati.i<.;  uri.  Ic  niakrs  witli  tlu-  \  it!  i,  al, 
llicil 

=  3;r;--^  -  ;■=(  ^ft*  -  4  sin  «  4-  ■"^"'/'^) 

~  2;-'(  I  —  cf.s  (4)(,T  -     >t  f  sin  ") 


ami 


/'  =  8/-  cos 


5.    Aqueducts.  -- The     ai|iu(liKt     ,.r    th,-     an    icnts     was     ,,f 
rci  tan-ulai    '-ntinn    ami  was   snnictinics   ,,)    \trv  l.ii-r   (limc!,- 
si.iii-.  as    I  iiiii|ianil   witli  the    Noluinr    (.t'ualer  tc  lie    com  rw<l . 
^I''i"'ir,li    1"  iii.Micrn   tiTiR's   tlicrc  arc  (  xampks  ol    icctani^ular 
sections,  it   is    ni.w    ninir    n  aial    ti'    make    iIkh!    i  m  iilar,    ci;},'- 
-^liaiiod,     s(]tiarr      uitli     a     .lia-;onal    xcilual,     ui      tia|ic/.M.lal . 
.\i|iiC(liicts   AW    ajsn   nnistiiHtcd    ,)|    (mis  whuh   a:  r  i  1  .nilaiia- 
ti.iiis  (it  till    I  111  if  and  ci'.Li-sliaiii  (i,  m  i,(  ijic  |.a|)c,'(iid  and  <  im  Ic 
\\  li>'ii    .1    'iiiaii   \oIiiiiic    ..t' water    is    lu    l.c    ,nn\c\a'd    and   w  ji,  11 
|.i()\  isi,,i,   has  tii  1„-  made    Imi   a  dclinitc  licii^lit,  as,  Im  (■\aiii|.ir, 
tor   a    man     ■taiidin;.;    n]irii;iit,    |ircfiTcnci-   is  ^ivcn    to   tin.-    i-.m. 
•<lla|ird   ai|iicdni  t 

In    till'    sections    sliown    hy    I'i^s     i;^    ti,    i  ?;   ,1   ^,,\\    |„. 
ol)SiTVL-(l   that    a  ri^c   of  tlic  watcr-lin.-  mar  th,     t..|.,awsr,    an 


Ani'li'i  CTS. 


:4i 


.i|i|)ici.  i.ihii'  iiuicaM:  ill  tlu-  v^ctt'.ci  iKTiniftiT.  while  tlu-ic  i--  ti" 
lirdportion.il  iiu  rci-t;  in  the  watrrw  i\  Thii-^  tlu-  mean  ilcjith 
,w)  and  llKictnic;  al-ciihc  mean  vclmit)  .-  of  fldw  continually 
diminish.  Ihf  ^i  /'r/.'n  concliisimi  ma\'  he  draw:  Ui.it  the 
di-ihar^e  iCji  is  imt  .i  niaxiniuni  wh.cn  the  pipe  run--  lull,  hut 
when    the   uatei-Iine    !■-    s:,nie   distanL'o    beloA    the    top.       1  iu 


;/ 


M 


»' 


^ 


■f* 


Vu..    133.  Fi'..   1.^4  I  I'i-    '35-  F"-    I"''  '■"'<=     '^"' 

dillerential  (((li.ition  detmip,;  thi^    po-itioii   m.iy  be  easily  lounc 
as  tnlh.w^  :  \\.:      1,  p.  22g): 

(Ji  A  . 

/>     .  =  />7'^  =  mi  =  pt. 


Th.eri  Inre 


^       I'  />• 


t  '  1 


Siiue   0  i"-  '"  1"'  ■'  niaxMniim, 

iiQ  =  o. 


1  heretnie 


r.) 


O    —  ,yt , 


or 


x.I\.iA  -  J  .  <//•       o 
is  the  equation  '-e(iiiired. 


I 


.4Qj.'i:i)ucrs. 


I 


Iftl 


IC    \cli)L-|t\-    l)t"  ttoVS     IS    \, 


and  therefore 


,im  —  o  —  </ 


>  \)u  a  niaxiniuni, 


—  o, 


.L       r .  </A  -  A  .  ,iP 


or 


I' .  ,iA     A  .  ,ir 


\.\.   I.      C.  I  ad, 11  .S, 


/Av^.  — Let    ll 


le   \^ettecl  i)erimeter  --uh- 


ternl    ,111    aiuMe    "    at     til 


I     eeiuri 


1 


,/n 


Kiice  |(ir  a  niiixniiinii  (ii.\i/i,ir. 


V 


d*>  .   "    1  —COS  ftf 


^111  "I  .  ,/H—.0, 


VlO.    I  !S. 


2«  —  ^"  ei>s  rt 


-)-    Nlll    «     --    O. 


H  —   ^OS     i-,  til 


e  \.[]\ir  ..I   "  sali-t\iin'   til 


1^  txiuatii  111. 


■I  a  /i/,i  \iiiti())i  Vilocitv 


.A'  .  "I  1 


t OS  f*  I 


{»  -  sin  <^)  = 


or 


«  =  tan  «, 


(1  ''    -::!?7''  -7'  i-^  the  \aliK    nl  n  ulmh  satisfies  tliis  (-(luat 
In   eireiilar   ai|iK-(liii  i  -    the   ,in;,;lr    "   i-;  nsnallv   ali 


ion. 


vviiii  h  insures  ,i  in  t.iiri 
Ihen.  ,iKo, 


lilt    J40^ 


■-pace  .i!io\  I'  f 


e  water-lnu 


/'        1.  i'"^')'  ;    ./  =  :'.i; J S ;•■•■';    w  .-  .  /,; 


A(^Uf-:DL:CTS. 


243 


pA'AMrii':   2.      A   Sijuivr  Sntiiiii  zfit/i  Vertical  Diagonal. 
—  Let   .1   side  iif  the   s([iiare         ,1, 
ami     let    ,1-    he    the    lellLTth    of  tlie  '' 

])nrtiiiii   of  the   sick'    which    is   not 
Wetted.        Tlieil 


and 


A  =  a-  -  ■ 

J'       :    4,/    -    2.1 


.Uld 


Ileiu'e  tor  a  uiaxniium  tlinliargi 


3(4,/  —  2x\\  .  ,/.!■  --  2\,r  —  \  j,/.v  ■—  O, 


.r-\ 


Tlierefore 


.r'  —  \  2(1.1-  -\-  2a'  —  O. 


X  =     (6  —  ♦'261  —  .  \Ha, 


and  the  deptli  hi  low   the  .ipcx  <i|  the  water-lino 


=  -;     =  .1274,/. 
V2 


I'"nr  a  iiiaxiiinoii  :  rloiitv  t'f  fli>'c 

—  xy^a  —  2  1  w/r  -i-  2\ti-  —    ^  yix  =  o, 
or 


4r7.r  4-  2</''  =  O, 


244  .-ini  f  ni  <:is. 

ami  tlicrcforc 

and  tlic  depth  of  the  watcr-Hnc  Ik-Iias-  the  ajicv 

X 

—  -        =  .4I4J,^. 


%^ 


m 


\ 


riXAMlM  I  ;.  /:';'•,;^-.^■/V(?/'(v/  ^, :  //I'l/ . —  Thi^  t'oriii  of  afjiu'ihict 
consists  essentially  otthrei  i>art-^.  a  hiwcf  [xirtion  hounded  \>v 
a  semicircle  of  radius  ;■  ,  .ni    upper   portinii   hounded   b\'  a  cir- 


--% 


V\t;.  1411. 

cular  arc  of  leaser  radiu".  ;•,  .  and  an  inteiMnediatt'  portuin 
liiiunded  h\  eiii.  ular  ares  dI  radius  ;^,  whiih  niet't  the  huM.-r 
and  upp^r  .u\  s  tanL;entiall\  , 

Tlie  tiepth  of  the  intermediate  imrtion  is  dttlned  h\-  the 
atude  cc  M'hicli  the  radius  (','',.  makes  with  the  lutri/' luta! .  and 
the  jinsitiun  (if  the  water-line  ././  is  defHH<l  h)-  the  angle  « 
which  (\..l  makes  with  (_'^( '_,  produced,       i'hen 


II 


r-i 


.md 


^HUEDUCTS. 

If  tlic  \\at(,r-!inu  is  (t/'aii-  /;/>', 

If  the  wator-liiic  toiiKidcs  with   /)'/.'.   W   -^  O.   and  then 


245 


t;V 


I  =  - '^  +  '■>  -  ('-,  -  >-,){.>■■-- '-,)  Mil  'f  +  :f 


'■., 


sin  2(f 


and 


If  r  is  the  vertical  di-tancc  between  0^  and  tlic  hi'^hest 
[)( nnt, 

-  =  >i  -r  ('•,  -  '-,)  ^^i'l  «. 
Also. 

'';,  —  ''i  =  (r,  —  r, I  cos  (r. 

If  the  uatcr-liiu  (C  1-  clow  A'/.',  let  0  he  the  an^Me  -uh- 
tendcd  at  (',  by  the  arc  /-'t  ,  .iiul  let  O ,C  ■=  x .  Tlien,  since 
li.pL    is  now  n. 


md 


^  sin  (0  —  0)  =  (;-,  -  r.)  sm  0 


^cos  (/9  -  a)  =  r  cos  ((»■  —  0)  -  (;-,  —  ;-|), 


two  etjuations  L;ivinj;  1  and  0  in  terms  of  "  and  the  radii. 

1  lie    are. I    (it      he  waterw.iy    is    now   the    are.i    up    to    lUi 

diminished   by  tlii'   a'  a   of  the  slice  between  A'/.'  .ind  (V".  and 
this  area 

=    -  sin  2,\  -f-  '■,'■'0  —  r .{r .  —  r,^  sin  0  4-  '     sin  2(H—  a). 


M' 


24') 


^(^LhDL  CIS. 


A=^-\- 


r.-tt   :  /•. 


'',  I' 


(r,  —  /'.,)  sill  a  4-  -' 


^m  2,t 


/    -• 


-in  2,,  -■   r  •<;>  —  /-  (/- 


.,)  sin  1.)  -|- 


-Ul    2(  H—  (t  I 


;t/-, 


-f-  rMir  —  (/') 


>'■.)  tV,  —  r.  \  sni  IX  --  /•  sin  r,'>; 


sin  2(14  —  ;/). 


1  lie  larLjor  diaiiutrr  i->  usually  at  tlu-  hMttuni  l.ir  a(iur(lucts 
fnit  ■ilni'ist  iiu  arj.iiil)   at  '.he  top  tor  srwcis. 


"he   dischai'LH'   tm' 


-ucr-    ma\-    he    eaK  ulat'.-d     h\-    ]?a/in' 


tnniiula,  luit  an  all.iu.nua-  of  Jo  ])ct  ecnt  shuuhl  In-  made  in 
iirder  til  make  ])r(i\-isi,  )n  t.  .r  depu^its  and,  where  the\-  occur, 
tor  uater-pipes,  eleetiie  cunduits,  etc.  Care  sIiduM  also  he 
taken  th.it  the  -section  i.,  suttkient  to  c.irry  .i\\,i\-  iIk'  w.iter  from 
the   hr.i\ii.-~t    r.un- ,md  h-oni  tlu;    luancii  tlr.iins  in  such  niaiir.cr 


th.at  the  w.itei'  in  the  se\\er  >li 


les  not  rise  aho\e  ,i  certain  le\el. 


.X-^-umiiV'  that   the  time  o|  tlow  in  ih 


e   sewer  is  three  times 


that    ipf  the   r.iintall    a\\\\    tli.it   tli<-    ni.ivimum    downtall 


dl 


■5 


\l\   htrc 


])er   scc'iiid.   B<'l.;r,uiil  lias   proposed  for 


the  disc!l.irL;e  ot  the  I'ari-^  sewer--  the  forniuk 


•S'  X   .  iSS         A  \ 


S   hcHU'    the   d 


r.una'e  .u'ea  in  .un- 


In  metric  measure,   .S  lieinij  tin    cl 


r.i'u.i-e  .lie. I  iii  hectares. 


.S"  X   .0239  ~  - '  •'W 

ill  hiMUi  h  lii.iiit-  .iMil  m  ^m.illi  r  -.'cst'ins  the  intlux  of  \\-, iter 
!'-  niiicli  more  rapid  .mil  the  time  of  lli  i\\  --hould  not  he  estimated 
•  It  iiii'ic  ih.m  twice'  the  diir.itioii  ot"  the  I'.iint.ill. 


Ii:iif( 


/( 


)i<.\fri..h  Oh  rR()\>.  I  YiiiHi  IS,  rrc. 


247 


I  *" 


N(>rF.--Iii  iK-^i-iiini;  -.rctuni-  f<ir  upcn  clianncl-^  or  aquc- 
diRts.  ti.itipliv.itLi!  j)irliiniiiar\-  cili^ulatiMii,  maybe  i;ciu.-rally 
a\ni,lc,i  l.\  c-inpiMyiii-  a  -rapliKal  iii(tli<"l.  SclcctiiiL;  a  i>r(i- 
MMoiial  -rctK.ii.  thr  \\att_r  ana->  aii.l  'vrttrd  porimctcrs  nia\  be 
i.htaiiUMl  lui-  (lUU'.ciit  (K'pth-^  ..f  watiT  aii.l  th.  i  oria -p' mdiiig 
nitau  depth--  pKittcii  t^  aii>-  (. onvenieru  ^cale,  Kepeatm^' 
tlu--e  opcratiMii-.  lor  ilitiereiit  ■-ectit.n'^,  the  nieaii-deptli  curves 
will  quickh-  iiidK-ate  the  be-l  seetK.ri  tn  he  adopted. 

6.  Formulae  of  Prony,  Eytelwein,  Beardmore  and  Tadini. 

A    earehil   -tud>-   of    (  he/,\'~-    eviierinient   on    the   ('ourpalet 

cut  .Orleans  canal  >  aiil  of  tu  eiit\--three  expennieiU.-  made  by 
l)Lit)uat  on  wooden  ihanneK  of  Muall  section,  led  I'ron.w  ui 
1.S04,  to  adopt  the  etiuatioii 

^'^-  =  ar-  +  />:■■=  w/, 

7V 


1  ,    I 

in  which       —  224.7:2.  ^  ami 


224- 


10^)07.02. 

About  the  year  1815,  l-.ytelwein,  takuiR  into  account  -^i.xty 
additional  experiments  on  the  Rhine  and  Weser  by  Woltmann, 
l"unk  and  Hrunin<;s,  proposed  sliyhtly  different  values  (or  a 
antl  /',  vi/,. , 


=41211.11      and 


Sg7  5.4.V 


■['he  exjiression  ;;;/  ha^  the  same  value  with  Tronx-'s  as  with 
l-.v-telweins  coeffu  ients  u  hen  the  velocity  is  about  ;^  tt.  per 
minute,  and  for  ,1  sm  di  dian-e  m  this  xeloutx  the  v.iriation  m 
the  value  of ////  is  also  small  and  ..t  little  practical  importaiue, 
lM)r  other  velocities  the  vaku'  of  ,;/.■  \Mtli  I'roin^  coefficients 
will  be  i^reater  or  less  than  the  \alue  uith  l-.ytelwem  -^  coeffi- 
cients accordm-  as  the  velocits  of  llow  is  -reatcr  or  less  tlian 
72   ft.    per  minute. 

The   tormula  with    i:>teluein  s  .oefticients  was   for   a    lung 


.4S 


lORMUL.h   ()/■    I'KONY,  lirrii/.U/IX.  ETC. 


tunc  u.cd  by  .n^i,u..r^,  ,hu1  u,,.  prH.-rrcl  a.  ^u,m,  th,-  nu,.t 
rcliahk-  results 

_  l-or  value.  Ml  ;.  cxcccl.n,^^  2(,  It,  p.r  nunuf  thr  trrn,  „;■ 
.s  s,nall  as  cnipared  u,th  ^:-,  an,l  may  Ik-  .|,s,v.;ar,ic,l  uitimut 
luudi  en,,r,        liii-,  r,n-niula  tlun  lu-o.nics 

d:'-  =  mi, 

an.!  thcrcl.irc,  a>  i  .-rditi-  f.  I'i,,iu 


•*. 


and  acicrdiiis^r  t,,  i;ytt  lucin, 

I 


♦  lilt  --   f  o  ^  i ' 


"//. 


>  /' 


\  lilt    .  y)-^  S  mi. 


liitcrincdiate  i.etu.vn  tli.sr  ,s  Ik'ardiiiur.-s  fcrmnla,  vi/., 

KK.)  \  mi. 
K.ittV-  ,1,    M.    \rnaMt  l,a.  suKUestcd  tl..-  rcl.iti.m 
mi  —  .coor  ^6.- Ti 

"T  mi  -   .00047'M,  if  a  Hutic-  IS  tin-  unit,. 

"'•■  ■'''■■^''  t'T'nula.  „„u  ..tw.ietc.  i.u-olvc  a  };,,,vr  rrmr 
as  It  is  assumed  that  ti.c  resi.sta.uv  du.-  f,  the  nm^luu-ss  ,,t  ,1,  ' 
wetted  surface  is  a  constant  quantity.  Ha.n,  s  .speriincnts 
I.av-  clearly  shown  that  tlu-  resistance  ,na\-  vary  betu.  en  verv 
wide  h.nits  .lependinK  upnn  the  nature  ,.f  the  materials  an.l  soil 
"liKh  form  the  bed  and  sides  ..(the  channel,  i-or  a  d,,  n  n-.\ 
u.-ic  J.annel.  in  which  the  slope  ,.f  the  bed  is  small,  appios.. 
niately  accurate  results  are  -iven  bv    ladmis  formula. 

:•  =  1)1   I  iiii 
I'T  s'>  »  mi.  il  a  metre  is  the  uniti. 


H.I /IN 'S    l<  )h;Ml  /  ,^- . 


349 


7.  Bazin's  Formulae. -iktwccn  1.S55  and  iS;,,,  I),,ri>- 
anil  l^a/in  carrir.i  out  ,1  rvin  irt  of  (.•\|Hrmirnt->  in  a  i  ut 
Icadini;  from  the  Hour-o;^'iic  tana!.  'I'lu'  eliaiincl  scttioti-  ucri' 
ot  (liltcriiit  fornix  and  dmicnsinn^,  the  si<lcs  ucrr  tarcil  uitii 
\vci..d,  tcnicnt.  h.un  a-hlar,  h-itk-^,  nil)l)l<  nia->nm_\,  .wiA 
caiih,  and  the   done  dftlu-  beds  varied  fnmi  ,'kii   to  .k  . 

I  lie  rt  ,iih-.,  t.ii  tin  111  laniMdar  and  tr.q'i  voidal  -cHtion,. 
scn-i!)lv  aiMxi  d  uitli  tin    ..  a  li  illation-  olitaiiud  tioin  tin       .rninla 


nil 


Init   with    .  inidar  and    -v,;; -diapcd    -ritlojis    tlu     laKul.itcd    arr 
alioiit    10  |),a  iciit   1(  -,  tlian  the  actual  if--ults. 
In  piailitc  it  1-  nio-t  i.on\cnicnt  to  takr 


V      ,  .  »  mi  -    c  V  mi, 
»  D 

1  K  '  '^ 

c  m 

.r  and  It  an-  rn.t  .  o'.stant,  lait  iia\c  \ahi>  s  d(|ii_'iidiiii;  uih.ii 
till  >  liar.u  trr  ..t  tin  i  lianncl  taics  and  Ix'd  l!a/iii  ^'ivt";  the 
lollowiivr  talijr- 


(  h.ir.u  itT  III  the  Wciinl  .Suri.icr 


Valur  "f  «,  I  ho  I'nil  brint; 


V<lu<:  u(  K. 


A  F.. 


\  Mrltr 


»     ] 

i  " 


'■mouth  icinrnl.  plane. 1  wimmI.cIi 

<'ut  nia<>i>iiry,  bricks,  plunks 

Riililile  iiiaMitiry 

Karlh 

M'juldrrs  (  Kutler) 


"•""■4" 

.IKKIH 

.OIMM,1)4|; 

D'KiOC  -« 

.<XII>||) 

a.ixMiijj 

>>'»«)73 

.OOOJ4 

.OiK  <>«) 

(X«)o«5 

.001)38 

.01)1)35 

IKOI2 

.uu<)4o 

t  M  ,1)7 

Tables  at  ilic  end  ..ftlie  i  li.iptir  },'ive  tlie  v.iliies  of  tlu- 
loefticients  />  and  .,  a  inr/r,  I)<.iiiL,'  tlie  unit. 

keviewiiij,'  tW  results  of  more  tli.m  7(X)  experinu  iit>  c.irried 
out  in   France,  luimiw,  the  I'niteil  States,  and  JJritisli  India. 


'-•50  G.4SGUll.l.H.r    -IM)   KlI'TER-S     !  Oh.Ul  i  ■>. 

itc, ,    upiiii    catiaU   .iii<l    rcxtaiiL^ular.    tr.iin/oi.l.i! .    ^-iiiiii.  in  ui.ir. 
ami  circul.ir  a(|ueducl-~,  nf  ditk-i-cin  ilmu-ii--i(>iis,  Ha/iii,  in    \>^<)~. 
.hill,  iiii  Poiiti  it  C/iii:iiSi, .-.  I  iliciui.L(i  tlic  formula 

157-6 


■fe 

'I 


or  V 


87 


t  m 


4  mi.   if  a  nutrc  i--  tin    imit). 


1  + 


I  ra 


I  hi-^  ('(maUon,   ,i;.;ain,   in.i\    Ix-  iiicisf  k  iiucniciul)'  urittcn  in 
the  foiin 

V  —  c  I  mi, 

and  1  .dill's  .It  the  i  nd  ^  i|  tlu'  i,,;]a])tci  i;i\!  tin  \.ilu(  ^  of  ,  for  tiic 
six  dilfiTL'tit  cl.issc-,  iiito  uliuli  l<a/in  Ii.i-  duidcd  ,dl  .  h.inncl-. 
thr  CoiT(>j)oiuliii;4  value-,  .  .f  til.  i  octlii.  iciit  .)'  hcniL,'  -ivcn  hv 
till    foIl,i\\  iiii;  t.ihli' : 


ClaH. 


Character  of  liie  Wciied  Surt.icc. 


I.      Smooth  reniriii.  planrd  wood ,, 

II        I'lanks,  bricks,  rut  masonry,  etc 

III.       Kiitiblc  niasiiiiry 

IV'.        I.     ill.  dry  rulj|)le,  eir 

y.       l-.arihrii  rhaiinrls  in  ordinal y  condition 

\'l.        I.aillitri  ili.iiMK-ls  .ir  rivi-i-.,  (.ifMiiiiny  cxci-plionajl 

rcsisiatiri-  ;  the  l)«'ds  rovrrrd  with  houlilcrs  and  I 

the  -'  !'•'  w  'f  h  gr.i'.'.,  I  I,  .       J    3,170         1. 


y,  the  I 

rut  biTnt; 

A  Foot. 

A  .Met. 

.loq 

.290 

.833 
1.540 

.06 

.  II. 
.46 
■85 

2.355        >  30 


8.   GanKuillft  and  Kutter's  Formula.      Ha/ins  j.  du   only 
fornuila  iisid   jn    !■  ranee,   but   in    lai^^laml,  (iL-iiiMn>.   .uid   tli, 
rniti-tl   St.iU-s  cnt^'inceTs   prcftr   tlic  tortnul,.  of  i  i.in-iidlct  .in.j 
KuttLT.  viz.. 

V  —  c  ♦  mi. 

the  vaiiii-  of  c  huinR  (jivcn  in  a  I  ,d)l(  .it  tlu'  vnd  of  tiic  Ui,ii.t<  i 


J 


G.-iMil  HI  i:i      IM)    K(TTHh:S    I'OKStLl.A. 
AUii  ihc  L'lcjtficiciii 


25  » 


a 


1        P 

n        i 
c  = 

/         p      n 
I  +  fa    •    *. 

1    ♦  m 

a,   1.   iuid  p  hciiiL;  ccrt;iiii  cm'^taiit'^  rind  ii  a  cocffioii-iit  il(|,iiiil- 
iiil;  only  (Hi  the    ii  iiiL;lnus--  ■  jI  tin    i  h.itincl  -.iiK  ■-  .imi  lnil. 

It' '.111-  unit  i--  . I  /.'.'/,  a        41.'';    1         1    NiiJ.    p        .00281. 

It'llu'  unit  i>  a  ii/<!ri.  a        2'\:   \         ' '.   p         oiu;;. 

1  he  iin:t  1)1  i III;-  a  tnut,  /,'  \  aric-  ti'nin  ■  .iS  to  .n;  ainl  tlu- 
tnlldwiii;^  tabic  ;;i\a-s  tiic  \aluc^  nt  //  wlijiii  will  be  tnuiui  of 
most  ii^c  in  practice : 


rhtiracter  of  Sidr* 


Atithority 


I'lant-d  timber 

S  nil  loth  cement 

A  mixture  nf  2  n|  tcmcnt  to  1  of  sanil. 
Kou>;ti  iilanks. 


'\2 


Ashlar  or  lirirk work .(J13 

(    III  V.I  s  on  frames        Oi; 

Kiihhle  inasoi;ry .     .017 

Kivers  anil  channels  in  vrry  hrm  Kf"^"^' ....    .oa        I    GanRuillet 

■'    prrfiit    order,    free    from  ile-;  and  Kuller 

trims  (stones,  weeds,  elc.).i   .025 
"  "  "         "    moilcr.itcly    K">'J    order,    not 

<|iiitc  free   from  detritus  or 

iveeils      '    .01 

"  "  "         "  liad    order,    wah     wi-eds    and 

<letritiis       035 

lOrrential  streams  cniuinbrted  vvilhdutriti^ 0% 

f.mals  in  earth  aliove  the  average  order    ■     .o2»5 

ill  lair  order 025 

"        "       "      lielow  ihe  averaKe  "fdci- ..     .0275    J-  J»ckion 

"        "       "      in  rather  bail  order,  overijrown  with 

weeds  and  covered  with  detritus,      .03 


( 


i 


I 


!  ! 


Ii 


S  : 


1  li'  iJiM'ii  iill\  '  •(  jii  I  i])irl)'  Mjlc-itm;;  tin.-  v.iluc  id  n  l>  tlui'  t"> 
till  lai  1  til. it  iluii  Is  Hi)  .ihs(»lutc  mt-asiirf  nf  the  r(>iit;lincss  of 
I  liaiMu  I  bods, 

in  '.bt, lining  tlu-  ii'iovo  ri'Siilts  {jaii^juillct  •\\u\  Kiittci  made 
.1  I  ail  till  stini\-  (if; 


«s» 


lORMll..^-    ,)/■    A/./.VA'/Vr;,    TITTON,  mC. 


11 


M« 


I 


II 

ri 


</'j  //v  /..xr"n,u,U.  ,./  />„,  ,.  „,,,/  y,W;,._ These  show 
that  .■  .Kpcuk  In.th  up..,,  t!,r  rou.^^hncss  and  the  sectional 
<iimens.,,as.  1  lu-  valur-  ,,f  «  and  /.  i,i  Ha/in'-  forDHi!,-'  v  iry- 
uith  tin  J,,,ract,r  ..'  tlu-  chan.ui  s,.l.-,  a„.l  I,,,!;  ln,t  u  Inle  ui 
M.iall  .ha.nvK  tlu>  inllue.u  ,•  ,m,,„  the  iImu  ..f  ,i„tere,u  e.  of 
n.u,;l.n.-.s  ,„nst  lu-  very  p^reat ,  „  ,.  .erta.n  that  this  influence 
'I'.nnu.he.  a.  th.-  seet.nnal  area  nureases,  an.l  that  u  uill  be 
ml   uh(  n  the  area  is  inlinitel\-  ^'reat, 

-^M  //v  M.<KSNn„u,ns  „f  //nui;,/n.rs  .v.:/  .\/>/,„t  uu  th. 
.Mi-M.s,pp,.  a  Mreau,  nf  very  lar-e  seeti.mal  ,irea  u,th  a.  bed  of 
\er\    small  sh.pe-. 

('  ///.v/-  ,/7.7/  aan,-;n,xs  in  the  re-nlated  cliani.eis  of 
certain  .Swiss  torrents  with  ex.  rptin„,,Ilv  .teep  slope.  an<l 
'iiniini_,  thr(.,.-h  .-xtremelN-  r-n-h  channels 

w/,  /•//<■  /://,■</,,/-///,■  .S^/<-.—  rhc  c..  muent  ,  ,lin,;.nshes 
..■^  the  slope.  /,  increases.  The  v.d.ie  of  <•  does  not  varN  nu.ch 
xMth  the  slope-  .,f  the  bed  ,n  s,„,dl  rivers,  but  in  laP^ye  rners 
with  sni.ill  slopes  the  \ariati..n  is  considerable. 

9.  Formulsof  Manning,  Tutton,  Humphreys  and  Abbot 
and  Gauckler.      h,   iS,,-.,   Manni,,,;  prop,  „ed  thr  i,.,nmla 

.    _  ,,;;/»/»  _.        ^~  111*1  \    i(  the  unit  is  .1  fn,,t. 


IT 


I  .1 

7-  —  ijiri* 


1       I 


"/'/  •'.  i(  the  unit  is  a  metre 


In  this  formula,  uluch   -ives  K«...d  results,  th,   oeffkiciit  ,/ 
has  the  same  value  as  the  //  in  Kutter's  fortnula. 
Ma/in's  and  .Maniiin^i's  form.d  1    .ire  identical  if 


»  »"        J  ^^  \  mi  =  ,,«/!/ i, 


i.e..  if 


F.X.tMri.FS. 


25:. 


I 


lU- an   indcpcndrnt   ii,,tlio,l,    Tuttiiii,  in   1S93,  lU.'duccd  the 
correspondiii;^  tormula, 


1.54 


in^i'. 


ti  bciii^'  aLjain  tiie  sanu-  a-^  m   Kuttir's  fi.riinila. 

As  a  rc-^ult  of  ohstTvations  on  tlir  Mi  —  .-Mppi  in  i.SO;, 
Iiuniiihro\s  ami  .\bt)(it  drdiucd  tlu  rallirr  ^  Mmplaatcd 
tbrniula 

•' "- .3-'*^73"«'''')^  —    ''^f^-'^;-'.   tlir  unit  hein-  a  toot, 


or 


'' =   ',(69;///*;*  -    .c>:ri4  •'.  tlh-  \init  IxinL;  a  metre-. 


In    tliis  fxprcssioii,  wliiili   is    of  cspcci.d    v.lUu'    tor    lar^'c 

watuioiirscs,  111'  is  tlic  ratio  of  tile  sectional  aria  t'l  tlie  t,fia' 
perimeter. 

(i.uii;kler's  tornuila-  for  tanal-.. 


<W/,  if  the  slope  is 


lid 


V  —  cm'/',    " 


~  per   i(Kx:)o, 
7  j>cr  looou, 


,md   I  lai/eil's  ('  iiiiiula. 


■4  ;,«/»/•. 

tlie   unit   in   e.icli    e    -e  heiii;;   a   metre.  lia\e    iioc   been    ii^ed    in 
practice. 

i;\.  I.  A  rhanii'"!  with  a  fiill  of  1  in  10,000  has  l>ri(  kwork  faces,  is  of 
rct'tan(;ular  si-i  lion.  30  fi.  ivi<li-,  atu.  is  to  cmivcv  .'cjo  1 11.  ft.  of  water  per 
!iecon<l.     What  tniist  l>f  the  (Icptli  >.f  thf  water  •• 

I.i't  .r  1)C  the  re(iuircil  di-pih.      I  hen 


,/  -  -<04^;     /■ 


.aid 


10  r 

+    I 


II 


i 

i  ■ 


;i 


U} 


1^ 


4 


% 

M 


a" 


W    t: 


^54 


i:.\.-tMPi.i;s. 


Also, 


2o.r  ~  -I 


'     lo  +    r  "  1 


/_IO£__ 

oooo       loo  '    lo  +  .r' 


This  (■(|ii;aKiii  c:u\  t)(-  hrsl  >olve(i  hy  trial. 
Let  .1—5  ft.     Tlieii 


'  3 


and  the   1  atirs  ^iv^•  I -^.^  .is  tijc  ci)rres[K)iuliiif4  \  ahic  of  <r. 
'1 'htTcliiie 


and 


^  '"    too'  *  -'-^'^  =  --'"^  "■  I"  '■  ^^'■•• 
(>        iDij?'  -  352. ^3  <  11.  It.  ]icr  SIT., 


whrii  |.-  too  yriMt. 

I.I  t    1      -  4  It.      Thi-ri 


40 
m  ^-      r.=  2.Sc. 

U 


ami  llu'   r.ilili  s  ^ivr  I  .'(1.4  ,is  iiic  i:i«rres|M)iirliiijj  value  of  c. 
Thcrcloro 

l2^'.4./4o 

T   =  -  —       V  -'.  1  j''i   H.    IICI    sec, 

100  14  ' 

anil  rj  :     ,Soj.       170.92  1 11.  ft.  |RT  s»M-.. 

uiiicli  is  iiKi  stii.ill. 

•j'liii.    ,   niip-i  li.-  Im  t«-i'i-ii  4  ..Mil  5  It.     Try  .1  : .  4.5  ft.     Tlien 


'4.5 


.VI. 


ami  tlio  corri'Sfioiiiliiii;  value  ol  .   is  126.8, 
Therefore 


'■        100  ♦    2.,     -   ---'3.18  ft- per  sec. 
£.'  -  'A*  '42  111.  ft.. 


which  isverx    nearly  ronert.     Hy  lurth.r  trials   tlu-  ilei-ih  tan   be  oh- 
tainiil  witliiM  .1  Irart'oM  of  an  iiicM. 


^i■ 


I'x.'iM  nil's.  255 

Kx.  :.  A  canal  in  earth  with  side-  -|.,|,nit,-  at  40'  is  to  rorivcv  lo.> 
(•11.  It.  .,f  wator  p.-r  sec.  at  a  velocity  ol  1  It.  per  .second.  What  is  the 
fail  of  the  canal.  .,ihI  what  arc  its  most  snitafiie  iliinciisions  .= 

A  —   100.     Tiieii  i-^cc  Tabic,  p  .t^;i, 

liMttoni  wiilth  .;:,   \  \oo    =:   ;.;5  ft.. 

depth  >  I  u.iici  .-11   (  100  —  7. J.;  ft.. 

me. Ill  ileplh,  m  ,\i,\    )  icxj  =  3,61  ft. 

My  the    I'ahic^  liu-  .  .ci,-,|,Mndiii)4  \''hie  of  ,■  is  1)3.  ;.      Th<-rclor<! 
I      ■-  o',,;,  t   v"i     '   /, 
1 


and 


31424 


l-.N.  3.  .\  leii^'ih  of  tiie  I„i  Ko(  he  cut  is  :n  c.  anp.ict  rock  Its  iiottom 
width  is  0.70  m.,  tlic  de(>tli  of  'he  w.iter  is  0.50  tn..  one  liank  is  vertical 
and  Iheotlier  slopes  at  :','  yX  t-,  the  v.rti.al.  Ii  the  fall  is  1  in  500. 
find  llie  mean  velocity  and  cpi,inii!\'  of  ihiw. 

The  widtli  of  sectior.  at  the  sinl.iec        .70    .    .5,,  t;,,,  y(,-  ^^-  _^  om  n- 

,  _  t 
•■'  -    2<-95  +  -"o)-  jo  ■--  '>.4'-5  s<).  ni. 

^'  =  .50  +  -70  T  .50  sec.  :')    34-        i.7jy  111. 


Therefore 
> 


.4125 
i-7S<t 


:.?4?. 


iiui  tlie  corrcsponditii;  value  of  A  in  the  Tables  is  .0423.      Hence 
."423  "  -■  -  1     -345 


500 
.51  2  111.  per  sec. 


.0216; 


Therefore,  also. 


.  ji  -•         41-5  -'I  I  :  c.tn    per  .<;pc. 

Again,  usinm  Hazins  formula  lor  the  tilaiiient  of  m.ix.  vef., 
^""«  ;  ♦    '         .512  f  14  X  .o2e65  =o".8i5  per  sec^ 


Mi 


1    *. 
>■    at 


anil 


J>        fxittoni  velocity     -   :-|J'nia«.)  -  o"'.  ;S9    per 


sec. 


2  56 


IX.IMFUiS. 


l'\.  4-   I"  another  letij^th  of  La  K.iclio  cut,  in  earth,  the  banks  slope 

at  43  .  ihe  bottom  width  is  0.3  in.,  .uid  tlie  depth  of  the  wattr  is  0.5  n,. 
Find  the  coellicients  />  :uid  c,  the  dihchariie  bciiiL'  .2112  cu.  ft.  |>pr  ■second, 
and  the  fall  1  in  500. 


-.U 


Tlii-reforc 

Also, 
Hence 


w  =   ^  I  1.3  +  .  51.3     -0.4  M].  ni. 
''  =  -3  +  -  1  .5  -^   I  ".714. 


1.714 
.2112 


=    -ijiT- 


=  o"'.528  per  sec. 


.528  =  ,t/. 2337 — =  — y -^M? — 

'        ■'■"500         I  /:'        -'-"500 


.02162    - 


.02162 


and  ,    -  24.4.         /,  ^  .ocjiOS 

uhi(  h  (  loscly  aj.rrce  with  the  lesiills  !;iven  by  llie  T.ihlcs. 

L.\.  5  Find  thr  (piantitv  of  water  conveyed  by  a  ch.inntl  of  trapp- 
zoidal  section  Inn  d  with  brickwoi  k  .md  liavinj;  a  fall  if  6  m  1000,  The 
water-surface  nidta  I  ■'.185  ft.,  the  bottcm  uidih  is  6.56  ft.,  and  the 
depth  of  tlie  water  is  4  >,     leet. 

1 

•'     -      ,<7"-5     +    6.56;     X     4.92     -:    33.SI3   S(|.    ft., 
/'    =    (>.•,(>     +     2     X     4.954    :--     16.468   ft. 


n   p:i 


Therefiire 
Hence 


3V^;i; 

III  =1  =  2.0;  I. 

I<)46S  '-^ 


IN 


0 


.It  =  ,    X  3.VS'jt'   -  "S.)   ^  =  <•  >:   3  7528. 


I'or  HI  —   2.O3J 

Razin's  Tables  Rive  <•  =  124.6.  and  then  O  =  467.6  <u   ft.  p.r  sec; 
M.inninj;'s        "  "     ,    -   1 28.*^-,  ;inil   then   (.'  =  482.6 

Knltci's         •'  ,    =^   130.4,  and   the!i  (J  =  489.36     " 

and  the  ditTeremes  in  the  three  cases  arc  i.ot  considerable. 


ItiLOUlY    i.-iRlAIlOt:    IN    TR/INSyERSE   SF.CTIOM.  257 

10.  Variation  of  Velocity  in  the  Transverse  Section  of  a 
Watercourse. —  The  discliargc  {(J*  across  any  transverse  sec- 
tion, of  a  w  itcrcourso  is  the  product  of  the  area  (.-ij  of  the 
section  ami  the  mean  veh)city  {rj  of  flow.      Thus 


Q  =  Ar. 


The  value  of  :■  for  channels  of  small  section  can  easily  be 
found  by  dischar^nn^'  int(^  a  suitable  reservoir  for  a  definite 
interval  of  time,  when  (J  can  be  estimated ;  and  since  .i  is 
known,  ,-■  can  he  at  once  calcul.ited.  Thi->  methml  is  imprac- 
ticable with  watercourses  of  larije  dimensions.  'I'he  pronle  of 
the  SOI.  til  m  mu  t  tiuii  be  i  ircfully  plotted,  when  its  area  can 
be  obtained  with  ,1  pl.minK'ter  or  b\-  the  method  ot  mean 
hu'-ht-.  The  \cloLity  of  llow  \-;;ries  from  jioii.t  to  point 
throu,L;]inut  the  -ection  in  ;i  mo-t  irre^nilar  manner,  ,'iid  its  value 
lias  not  been  h.xed  by  an>-  sJUL^le  law.  By  ii-in,  a  meter  or 
'f,ui"e  the  \elocit\  iii.u-  be  measured  at  a  lar^e  number 
of  i)oinl-,  .uid  ill  tin-  m. inner  the  mean  velncicy  c'l  and 
the  ma.ximn.'i  \elocit\-  (.•„,„!  can  be  very  aj-proxiniately 
determined.  1  he  \elocity.  however,  v.irie-  -.  iiuk  h  .mil 
d(  piiuK  >i,  l,ir.;el\'  ii])"!!  the  conditions  under  which  the  How 
t.ikes  jihue,  th.i'  it  -(.(.nis  hopeless  to  expect  th.it  the  coi.-'uli- 
caled  l,iw  (if  \eliK  it\  distribution  cm  be  expressed  in  .'i  :^^ener.d 
formula.  The  niimer.>ns  ex])eninents  ot  Ha/.in  on  the  I'.our- 
"■(iiMie  i.iii.il  and  on  tlie  Seine  .iiid  S.ioiie.  'A  t'inininL;li.iiii  011 
tl-.e  (i,in,L;es  cinal.  ai'd  of  lluniplney-  ,ir,d  .\bb<it  mh  the 
Mississippv  .lb  -o  to  pio\e  this  ,111(1  ,it  the  s,ime  time  throw 
much  liL'.ht  upon  the  whole  subjea.       It  h.is  been  shown  th.it  the 

i.itio    ' '""'    diminishes   as   the   resist. nice  of  the  sides  and   bed. 


if 


lie  r.itio, 


which  It  mc.isiired  b\-  th-.'  expressK.n       ,.  iiu  re.ises. 

for   ex.unple,    is   .ibout      S;    in    .1   (  ii.mnel   with   ,1   ver\-   sniootli 
surface  and  t.dU  lo  .dxnil   .  ;"  w  hen  the  i  h.iimel  is  i  ut  throiiL;h 


^5^        yi-UKjiY  r.-iKi.-iiu 


'A    /.N     IRASSriiKSI-:   SlA.IION. 


cattlK  As  tlu'  surf.iLc  ixM^t.nuc  (!immi-.lu's  tlu'  \-,iliic  of  '"','' 
tends  t,.  iHT.mir  vrr)-  s„v,!l  :,.!■!  ii!ti;ii;itrl\-  /.n,,  ul,,lc  tin- 
ratio  '""■  tcii(U  to  \h-^:wc  1,11  III.  H,i/m  tlu-rrlorc  express  il 
tlic  rcl.itioii  bttuccn  ,•,„,,    am!  ;    in  tin-  ri.ini 


=  !+/■ 


.jilt  I 


111  wllKli  the  hiiutiMii  /•  (  -,      v.iiiislirs  with 

.\  special  ease  is  Ha/in's  empirnal  lorimila, 


i'> 


*  max.  . 

=  I  -r    /^ 


=  I  +  A'  •»  /■ 

A' 

=  1  +  --.     . 


(2) 

(41 


the  vahies  of  A  anil  r  t.eint;  t;i\en  hv  the  Taljles,  and  A'lieiiv 
a  eoelfn  lent  dejiendinL;  upon  'lie  Inrni  .it  the  sction  and  the 
eond-tiMiis  ,,t  il,,u.  1  .ir  example,  if  ,  ,„,,  is  tlu-  inaxiiiiuin 
.inrjiin'  '.tli'iity  till"  a  l;i\(.-ii  sfttKui, 


!P-^ 


--^  '  -f3f'-:. 


nm 


Vi^'    • 


•     (5- 


In,    ,1   •A,aeii..ii|se    ot    L;ieat   width    as    lonipaied   with    tlledrpth 
and 


V 


I    T    -\^.4 


'  >ll/ 


v.'- 


(6) 


tor  a  channel  ol  le^futed  (hnun-^i.  ms,  as  in   oidinar\-  practice. 
.\i;ain,  if  ,■■„,.,,    is  the  iiiaximuut  \eloeit\-  t.ir  the  whole   sec- 
tion   cf.iKh    a   ihami.l,  m\\    iI    ;,,    \^  kW  nicnn   \  el.  ,cit\-  al^n..- 


ir. 


yt:/.()(jriY    r.lh'l.-UIOS'   /V    /R.-IKsl.'FRSf:  Sf-XT/ON.  259 

the  vertical  in  whiJi  tin;  in,i\miuni  vc'.outy  lies,  ti<;.-n, 
iippi'oNiin.itcly, 

•■max.  =   '•<„+   10.9  t'/// (7) 

in  v.hifli  //  is  tlic  (l.'pth  ni  tin-  water  on  tlu-  vertical  in  ([ues- 
ti"ii  lit  .^  metre  is  the  unit,  the  values  o!  A' in  the  three  last 
Inrniiil.e  are  jc,    14,  an'l  6,  respective-U-. ) 

I'll'- eliannels  of  mean  llimen■^l(lns  lVon\'  lias  suycsted  the 
formula 


I' 


i  i   ; 


/•/^  Hh  <''m»x. 


(8) 


ill    the    unit  i-.  a  n.etri\   ^ul)-^tltute    -V  37  for    ~.~X.  .iini    3.  I  5    for 
iu.;4. 

In  the  ^ame  e.ise   iMihiiat  L:i\e-- 


fj 

--  4 


*''  max.  "T  ^'/ 


(9.) 


in  which  ;■,  is  the  veh"icit\-  at  the  bottom  (.f  the  eliannel. 

iMir   values    of  :■  ,„.„    u[)   to   about   II   or  IJ  It.  .3.5  m. )  per 

srcoinl  the  c.ik  ulated  wihies  of  the  r.itio        \ar\'  but   little  from 

till-  a\er.it;e  \-alue  .S,  a  ie--iilt  which  h.is  bee  :i  \i-nfiei!  in  cert.'iin 
-jieiial  experiment-.  it  1-  therefore  con^idereil  suftlcieiii  to 
take 


If     t  : 


^  .8, 


;uul  then,  b\-  ecj.   (9), 


f^-^.6.       . 


When    the  uater   is    of  i;reat    depth   the  ratio 
.75.   and  to  .IK)  11  the  i)ottom  1--  imcnil   with  leeiN 


(10) 


(':) 


alls    to 


'  iH 


y;i 


•  < 

Li 


MICROCOPY    RESOIUIION    TEST    CHART 
iNSI  OPd  ISO  TEST  CHAR!   s 


1.0 


?H 


f  3- 


I.I 


1.25 


1.4 


iiii  2  5 

2  2 

I  2.0 

1.8 
1.6 


^      'IPPLIED  IfVMGE     Inc 


a6o        vriociTY  /'./AV ///')'    .V  ik.-i\sii;kse  sf.chow 

Sonnet  has  tlic()ii.'tii.ali\-  ikc!iui.''l  tm-  \\,ttt-i\iiurscs   ot  ^rcat 
width  the  relatid.i 


'''/: 


(12) 


so  Ih.it  if 


■.n.,x  .  tlii-'ii  '■/,  =  '■ 


•*. 


^"    -.n.x   •    I'll-'"    '■/,  =    i!''m..v 

I'Or  a  imig' time  it  was  mipposcci  that  the  niaMinnni  \  ilni.  ity 
•^'n.nx  '  ^^''■'  '"  ^'"-'  ''"■''■'  '~^"''"-<-'.  ''ii'l  it--'  \Mhit,-  was  ik.'tLinuiK'il  !iy 
obscrvinj;  tlic  time  m  which  llnats  ])as>cii  between  two  trans- 
verse sections  at  a  specified  thstaiue  apart.  I'.Nperinuiits  lia\e 
now  demonstrated  that  this  maxinnim  \elocit\- is  at  some  ji.'iiit 
l)elou  ,  althoii^'h  in  ijeneral  near  tin-  free  surface,  and  th.  tloats 
will  not  ^'i\(.'  the  jiroper  '.ahie  ot  the  maxinunn  \elocit\-  mdess 
the\-  are  suitabl)  ■-iii)mer;4ed.  It  has  also  been  found  that  the 
depth  oi  this  point  (^f  ni.iximuni  \cloi  ity  increases  as  tlie  ratio 
of  the  wiilth  to  the  depth  of  tlie  waterway  tliminishes,  ainl  ma_\' 
be  as  },'reat  as  ,>;/<■  /////r/ of  tlie  depth  of  the  water. 

On  aiu'  horizontal  line  at  ri^jht  angles  to  the  a.vis  of  the 
lii.mnel  the  velocit)' iliminishes  w  ith  tlu-  depth  of  tlie  water, 
is  jjreatest  towards  the  centre,  and  diminishes  at  an  incre.isinjr 
rate  on  approachintj  tlu'  sides. 

1  he  experiments  of  I  )arcy  and  Hazin  liave  shown  th.it  the 
air-resistance  is  not  the  nio^t  imimrt.uit  factor  in  cau■^illi,'  tlie 
variation  in  the  velocit\-  throughout  the  section.  With  a 
jj.iufje  they  determined  the  velocities  ,it  a  number  of  jioints  in 
the  cross-section,  and  plotted  the  corresponding,'  e(iual-\elocit_v 
curves : 

(<f)  l"or  a  i/i'.\i</  woodin  I'ij'e.  of  rictan^,'ular  --ection. 
running,'  full  t  l"\n-   14'    '. 

(/')  l''or  an  f/'iii  womlen  liiantul  runniti;;  //<///"  full  .md 
formed  by  removint,' the  upjier  side  of  the  pipe  in  iii)  i  I■'i^J. 
142). 

The  curves  for  the  pi|H'  ,ire  approximate!)'  rectangular  ,\iu\ 
par.illel  to  tlie  siiles  of  tlie  pipe  I  lie  ilisch.ir'^e  in  the  open 
ciiannel  is  slijjhtlv  greater  than  one  half  of  the  pipe's  disch.irj,'e. 


yEl.OaiY    l-.-IKI.-lllOS   IS    IK.-ISSyEKSH   SECTION.  261 

hut  iherc  i-^  no  similarity  between  the  ecjual-velocity  ciirvc'^  in 
the  two  cases.  In  tiie  open  eliannel  they  become  more 
eUiptical,  tend  to  close  at  tlie  centre,  and  cut  the  free  surface 
obhiiuil',,  the  an^de  of  incidence  becoming  more  and  more 
acute  towards  tiie  centre.  The  curves  are  also  at  a  greater 
•  h^tance  from  the  centre  than  tin-  corresponding  curves  in  the 
j.ipe.       1  his    vciy    marked    moclihcati.  m    in    the    torm    ot    tlie 


zr^.z:-^ 


Fill.   i4t. 


Ki'..   142. 


velocity  curves  is  due  especiall)-.  in  Hazin's  opinion,  to  the 
absence  of  tlie  upper  boundary  and  to  the  ronseijucnt  practical 
imi)ossibility  of  an  absolutely  constant  cross-section.  Mtldies 
and  otlier  irre^jular  movements  are  produced  in  tlie  surface  and 
;,'ive  rise  to  correspomlinj,'  losses  of  ener^;y  and  velocity. 
.\ctual  experiment,  too.  has  shown  that,  even  with  a  stront,' 
wind  blowing'  down-stream,  teiulinu.  as  mi;,dit  be  supposed,  to 
c.iuse  an  excessive  surface  velocity,  the  m.iximuni  velocity  is 
still  at  some  pf)int  /'</<'«'  the  free  surface. 

I'or  any  ^,'iven  vertical  in  the  section  it  appears  to  be 
.ipproximately  true  that  the  vel.)city  at  about  ///nr /i/t/ts  of  the 
total  depth  is  sensibly  the  w, ,»«  velocity  for  the  whole  dejUh, 
•m.l  that  the  dilTerence  Ix-tween  the  maxinuim  and  bottom 
velocities,  viz..  :„,....  -  J>.  increases  with  the  rou-hn-  -  .."d  li.s 
between  ii„„.  and  ir'„„.. 

In  a  semicircular  channel  of  radius  ;•  the  eipial-velocity 
curves  are  circular,  Fij,'.  143.  ;vn'l  concentric  with  the  bed,  the 


2<')2        iT.iociTY  y.-iKi.-niox  i\  //\'./.\.sKi/<5£  .sfcr/o-v. 

velocity  :■  at  ll'.c  dist.iiKc  _)■  iVwin  llic  centre  hciiii;  given  by 

-•  =  •  V  -     ^,  ~y\ 

7\.  beini;  the  velocity  ;it  tile  centre. 


1^ 


m 


t:'- 


\'\'..  113. 

rieneralh'  speaking,  tlie  <i|\ial-\'elncity  curves  are  approxi- 
mately of  tlie  same  fi>rni  as  tlie  profile  "f  tiie  section  ( I'if^s. 
14;  to  I  4'    .  au'l  tiii-  is  e>i)euall\   the  case  near  the   sides  and 


■'4.    :>■ 


I  14. 


■.-. . '>;n-J>- 


Kli:.    1  It. 


Ii.:    1)6. 


bcii.       1  he  curves  at  tlie  bottom  tio   not  always  re.ich    the  sur- 
face, l)ut  sometunes  cut  liie  sides. 

Again,  ex|)erinu'nts  indicate  tliat  the  l.iw  of  velocit)  ilis- 
tributinn  along  any  vertical  in  the  section  may  he  represented 
by  a  parabola  of  the  2d  degree,  with  its  axis  horizontal  and  at 
the  same  ilejith  as  the  point  of  maximum  velocity  Defontaine 
in  an  e\i)eriment  on  .m  .uni  of  the  Kiiine  deiluced  tor  the 
vertical  .it  the  centre  of  the  current  the  analo{^ous  law 


II        .\.X22  2      ■  .oTh')!'-', 

II  being  the  \elocitv  at  the  depth  1 . 

U  the  umt  is  a  metre,  //   -   l.2()C  —  .252_/.) 


('3^- 


yiii.oan   i-ARi.rnos  is  /a'.^-vm7:A\s/;-  si-ciiox.       -j'-j 

riic  followin--  theoretical  invcsti^Mtioii  ot  the  velocity  curve 
i^  based  on  the  a-sumptions  that; 

,,n   The   watercourse   is  of  very   '^n^nt.  width   as  compared 

W  ith  the  depth  . 

(/'I    The  watercourse  is  i.t"  sensibly  unitorm  depth; 

(,)  The  fluid  particles  flow  across  a  transverse  section  in 
sensibly  p.u-allel  hnes; 

{,/)  A  permanent  ref;iine  has  been  otabhshed  >.>  that  the 
pre--ur.-  is  di-tributed  over  the  section  in  sensibly  parallel  lines; 
,  j  Ihe  resistance  to  the  relative  flow  of  consecutive  fluid 
filaments  is  of  tin:  nature  of  a  viscous  resistance. 

l.cl  l!-,    147  r.^present  a   p.^.tion    of  a  vertical  h.nijitudinal 


*"-  —   J.. ' 


r- 
I 


i   '- 


Fii;.   147- 

section    of  the  streani   intersected    by   Mvo   transverse  sections 
//,'    tV    /  bein;;-  the  distance  between  them. 

Consider   a   thm   laver  ab.d  o(   thickness  dy  and   wi<lth   /', 
bounded  bvthe  sections  J/.\  (  /',•  and  by  the  planes  a,t  /'C 
at  depths  ;and  v  ^  dy.  respecticel>  ,  below  the  free  surface. 
The  f.'.rces  acting'  upon  the  layer  ,n  the  direction  ol  mot.on 

'I''*-"-  ,       11  1     .1,. 

(I)   The   pressures   .n,    the   ends   av.   ,-/,    which   .- .!.-,!> 

neutralize  each  other. 

{2j    The    component    ot    the    w.-i-ht    =  wbl.dy.^m 
'a'Hi.dy.   I  beinn  the  slope  of  the  bed. 

(,,"The  visc.us  resistances  ,.n  the  lateral  faces  ..l  the  laser 
unci.;    consideration.      These    are    nil.    since   in    a   stream    o 
in.letlnite  vvidth  there  will  be  n-  relative  shdinK  between  al'ai 
and  the  vertical  faces  on  each  side. 


i'i  ', 


jfn        ymociTY  y.-iKi.-iTiox  t\  rh\-i\'sy/iKst:  section. 

(4      Tlic  \iscous  resistances  alonij;  the  pl;uic>  (;</ ami  /'c. 

I  lie  frictioiial  resistance  to  distorticin,  i.e.,  tn  shearing.;, 
alwiiL;  sucli  planes,  is  fouiul  to  be  ijrojjortional  to  the  shear  per 
unit  '  if  tune,  and  is  iiuMsiireii  1)\-  the  -hear  per  unit  nt"  arci  at 
the    actual    r.ite   of  shearini;        I  he   coefficient   of  %iscosity,  or 

shear    ]ier    unit    of  are.i 
she.ir   per    unit   of  time' 
and  delines  that  quality  of  the  iluid  in  virtue  of  uhicli  it  resists 
a  e.ianj.;e  n\  sh.ipc. 

.Adoptini;  N'a\ier's  h_\pothesis, 


sin!pl\     the    \isCosity,   is    the    (juotieilt 


the  \iscous  resist. mce  aloilL,^  n,!  ~  —  k/'l 


du 


k   Ww^   till-   CMcftuii'iit   iif  \iseosit_\.  and  ,7  the   \el<icit\'  at    the 
depth    !'.        rile    si^r,!    is  net;ati\e   as,  sinci'   ti   increasrs  with,    j, 

Is  po-itui,',  and.  .It  the  s.mie  tiiin',  the   .u  tion  (it    the    hu'ers 
ay 

al)o\e  (/,/ is  iif  the  ch.ir.ictir  ot' .1  Kt.irdation. 

'Idle  \iscous  resist. iiu  e  .doni;  /'<  =  kl't-    -\hl>l    </l       ) 


hen,  .i-.  the  motion  is  unifdrm, 
iiu       .  .  .du 


i'u 


I:-' 


,  ,■     ,         I ,  ,"u        , ,  ,'iu    ,    ,  ,  ,(tu 
wbli .  dy  -  kbl  -.    -t-  /•/'/  ^   -f  khl-r-Jy  =  o 
dy  dy  dy  ■' 


d'U         Ti/ 


Intct^ratini^  twice, 


u  = 


:i7 
2/- 


,.  .!_  -. 


a  .111(1  7    Ining  constants  of  integration. 


(•4) 


yF.l.OCITY    yARl Alios   IS    TKAS^yhRSE    SF.CTIOS.  265 

It  is  evident  tiuit  ;-\  is  the  surface  velocit)'.  i.e  ,  tlie  value  of 
u  uiien  y  —  o. 

The  ec[iiatii)n  111. i\-  be  written  in  the  form 


(15) 


Thus    tlie    \cl()city   curve    is    a    parabola 
ha\inL'  a  In  in/i  int.il    axis  at  a   lieptli    )    ^   ~  . 

bel<i\\-     the     fi'ee    -.urf.ice.        1  his    i-,    aN<i    the 
(.le[)th    iif  the   tilanient  nT  maxiinuni  velocity 


l;r 


».,-•'.+  ,.(,/       •'. 


'Wl 


r- .    uf)) 


Fii;.   14b. 


I  leiice,  b\-  equations     14    and     16), 


-.7 


,1   I' 


)'|-'. 


(\7) 


\ 


T.et    7,,,    be    the    "mean"   velocity    for    tlie  uholc    ilcpth    //. 


Let   ;  .  be  the  nnil-depth  velocity.       Then 


•It 


•  m   — 


-g^(//^-3/'^'+3n.   .      •      •     (>«) 


and 


t  ~~     BUI,        -•/(•':;  ,' 


.     .     .     .     (>9) 


260  yULOUTY   yAKlAllOS  l\    I KA.\.si.LHSl-:  SlirjlON. 

1  I  dice 

2^k 


(20j 


a    rc-^iilt    npiiil    wliicli    !  lumphrry-;    ami    .Ahhut     li,i\r     l)a^cil    a 
rapiil  iiKtlii"!  lit  L^auLjini;'  ri\frs. 

Let    r      \fv    tin-    liottom  \cliicit)',  i.e..  the    value   ul    ;   when 
y  =z  ft.       Then,   b}-  equ.itiiiii   iiy), 


''a   —    "^  max.  t/-*^  -*  ' 


ami  tluTefdi'c, 


'W! 


-  'a  =  ^^*/'  —  y  f  =  '^'-    -<jpi)"se. 


(21) 


Ai-i-ordiiii,'  ti)  H.i/in,  r'^j,.  —  :■„  is  sensibly  cim-tant  and  is 
aiiprMNiin.iteK  (liii.il  tn  36.  3  ♦  ///  (  =  20  \  hi  W  ,i  metre  is  the 
unitt,  Tlui^  '.he  f,'ener.il  eciu.ition  1  I  ;'  (jfthe  \  eh  leity  cur\e 
becomes 


"='-'max."  .1*^'-.5  * 


-  y: 


'^'e:;j 


(221 


1:=- 


This,  knnwn  .is  H.i/in's  tnrnml.i.  a^'rees  wall  with  tlic 
experiment--  "ii  artilRial  ehanneU  ami  <a\  tlie  S.ume.  Sein.e. 
Ciaroiine,  .md   Rhine.       It  w.i--  tMuml.  m  i^eneral, 

tl,at  l.!2::l  —    l.|-  111    the    Khni.-   ,it   I'.a-le   ami    r.m^'ed  finm   i.i 
t..   1,1',  in  tin-  'itlur  i  h.nniels; 


?6.  xli-  S' Id  ,  , 

^,-   -  ,.  ..    rantied  fn  111  1  ?  to  20; 


yhLouiY  I'ARiAiios  IS  /a'.-/a:s W;a:j^/;"  section.        :''7 

)•     1 


/'   ~  3 


in    some    aititK:i.-il   eh.iniK-N   ami   in   .'thci-  r  uiL^'cd 


frcini  o  ti) 


lei 

^  f 


These  last  results  are  not  in  .uxmiiI  with  the  Mi-si^sipjii 
measurements. 

In  the  ea->e  of  a  reetaiiL^ular  channel  of  -ik  li  width  that  tlie 
iiithience  of  the  titles  on  the  llow  ma>-  he  (lisre.Liarilcd.  the 
nie.m  rachus,  >//.  may  he  sulj-^tituled  tor  /'  ami  tlie  mean 
vehnity.  ,-■„,,  i>  seii-^ibly  the  -ame  a-  the  mean  \eloiity.  r  ,  tor 
tlie  whole  >eetion.       Hence  oiuation  1:2     may  Ik;  written 


^'    'max. 

I'    ~"         V 


III!  !] 


!v  -  y\- 


s^-Vv-i-iJ- 


IV  -  y\' 


(23) 


the  value  of  /■  lieiii!^  LM\en  !)>•  the  Tahle-^. 

lilaiiuut   ,'!'  Mii.xni.inii    \\!,-ii/_v  ni  ///<    .Siii/an.  —  In   this 
case   y  —  o,  and  eciu.ition  iJl)  becomes 


2  k 


/'\ 


(24) 


:■■„,„   hcin<r  the   value  <>f  .•■„,„.  ulien   the   maximum  velocity  is 
in  the  surface. 

M(|uation  (_22)  also  beconie-- 


-T-:!" 


«=   -max.-  3^"'- 3    V ///■,,.        .        .        .        (25) 


ALrain.  b\  eiiuations  (  iSi  antl  ^24) 


KV 


••'ma..  -  '•*  =  2l-^''  -    3('' m...  "  '•-.)• 


a68  yil.Oi.ITY   yARI.-^nOS  is   TR.-tNSyERSE  SECTION. 


or 


i:--! 


"•. 


•'7''  -I-  t'z 

-'    max.    1^  '  * 


(26) 


a  result  rilrcady  referred  t<i 

Jn'i/iii/t's    l-orniul(C. — H'iilc;ui    a'^-^iinic-^    tliat    tlie    \elo(.ity 
y,  __    .Wt  .y    cur\'e  is  _L;i\eii  by  the  e<iu.iti<in 

It  =  J  —  />>-  +  Cy     .      .      (2j) 

a/wrc  the   ])Miiit   of  niaxinuir.    ■  locit\-,  am! 
/'t/cii'  this  [jiiint  b\'    the  filiation 


u  =  /;  -  />y.  .     . 

When    y  —  o,      u  =■  r,  —  A. 


(28) 


When  y    --    r, 

/'  -  /:y'  -  ■■,„„.  =  .1  ~  /n-^  +  cv  =  v,-By-j(-c  /. 

Thrrefiire 

J)-v 
C  —  —    — 

n<iileau's  experiments  led  him   t'^  infer   that   the   difference 
^'-    •  m.ix.  I  —  ■^'' ^ ' ' )    ''^    sensibly    on-^t.int.       1  )e^i;_;natin;_;    this 

differenec  by  (/.  so  that  D  =  '''m.>t   +  '''  '''"^'  i'  =    i-^'  IJoile.iu's 
iiiuations  lux  I  lUie 

Jy  V         'ma:..  "  ''.  +  A'  ^ 

u  =  rs-d[y]-  -^ j^  .      .      .     (29; 

representing;  the  ctirvt-  MM .  .  and 

"  --■m..x.+'^—  '\y) (3"' 

representing;  the  curve  .l/.J/,. 


T.-tliil.S   Oh    l'.ROSlO\   A\'l)    I^'ISCOSITY- 


2O9 


II.  Tables  of  Erosion  and  Viscosity. 

lAHl.l.    INDIC.MING     rHK    \  11 .' H  1  r  1  KS    AHoVK    WHICH 
EROSION    COMMKNCES. 


N;tlure  of  Ihc  Channel  Hnl 


Soft  and  cLiyey  soils. 

Rich  clay 

Kirin  s:iiui 

Gravel 

Broken  stone 

Soft  schist 

Str.ititied  rocks 


Met.       Feet. 


I" 


Mel.       Feet. 


-15  -40 

-yy  -q*" 

.  (10  1 .  97 

.  I  22  4.00 

1.52  5  00 

2.2J  7. 2* 

275  9.02 

ll.irl  rocks 4-27  M'-o 


■  21, 

.40 

..,6 

1.23 

1.86 

2.27 

3  •  69 


.36 

I. 51 
3»5 
4.03 
6. 10 

7-45 
2. 10 


Met. 


.08 
.16 

■31 
.70 

•94 
1.49 

1.82 

3  14 


Fert. 


.2(> 

■  5- 

1 .02 

2.  30 

3.08 

4. go 
6.UO 
10.3 


•tabu:    ok    VISCOSITIE.S    {.Enrttt's  Systtm    ot  Inils). 


WATER. 


Temp, 
(lent  1 


5 
It) 

«5 
20 

25 


Viscosity. 


.OlSl 
.0154 

■  o'33 
.01 1'l 
.0102 

.0<K)I 

.  I  08 1 


'[.'■"'''        ViMosily.  I 


3? 
t" 
4~ 
50 

<)0 

bo 

90 


.0073 
.00(7 
.txrfii 
.<K)56 
."0<7 
.0036 
.0032 


MKRCCKV. 

Temp. 
(Cent.) 

Viscosity. 

Temp. 
(Cent.) 

0* 

.Olf.9 

-,  i;° 

10 

.<il62 

340 

IS 

.olcO 

09 

.0123 

I?4 

.OIOJ 

"97 

.0103 

249 

.o*K)f.4 

Viscosity. 


.  I  o<>  I S 
.00897 


Tlir  \i-ni-it\    1-  -ivfii  by 


.OliS3 


ncoriliiiL'  ti  I  Mi\rr, 


.111(1  1)\' 


■'"'    "         001  ^i,  iucnrdini;  to  .'■ilottc; 
26  -f  / 


!  W'uv^  tile  tcmiKiMturf  i  c'titi<;raclc. 

12.  River-bends. —  I'lic  tollowint;  explanation  i-^  dm-  to 
I'lofcssnr  James  ThoniM.n  Inst.  Mechl.  Kn^'s.,  iS;.,;  I'roc. 
Royal  .Soc.    JS//"'       l"    ri\x-rs   tlowin^;   in   alluvial    plains,    the 

•  N     n.   The   viscosities  are   in  C.   (i.  S.    units.       lu  tciiucc   1      '  >• 

units  and  centiKra.le  degrees  multiply  by  2.0431. 


»70 


Rn-i;R-HENns. 


Ui! 


fur\'aturc  ..f  tlic  uindinijs  which  ah-caily  exist  tends  tn  increase 
owin^  to  tlic  scourinL;  awayol  material  hum  tlu;  uutir  hank 
and  to  tile  deposition  of  detntii-  almiL;  the  inniT  hank.  'llu- 
sinuos'ties  often  increase  until  .i  looj)  is  t'ornied,  with  oiil\  ,i 
narr(A\'  isthmus  of  land  between  t\\o  encroacliin^;  l)anks  ot  a 
ri\er.  l-'in;dl\-  a  cut-off  occurs,  a  short  ])assa^e  for  the  water 
is  opened  lhiiMiL;h  tlie  isthmus,  and  tlic  looj)  is  separated  from 
the  ri\er-i  ourse,  takini;  tlv  f>rm  of  a  horseshoe  shaped  laL;i>on 
or  suanip  The  nrdinarv  supposition,  that  t!ie  w  ater  alw  a_\'s 
tends  hi  \\\"Vv  iMivvard  in  a  straij^dit  line,  rushnii;  aj^iiinst  the 
outer  hank  and  wearini;  it  away,  and  at  the  same  time  i.ausinL,' 
deposits  at  the  nnier  hank,  is  i.orrett.  but  it  is  \  er)'  l.u'  h'oui 
beini;  a  complete  i  xplanation  ot  what  takes  jilaci  . 

When  \\ater  tlnws  round  a  circuhir  cur\e  under  the  aiti^n 
of  L;ra\'it\  only,  it  takes  a  inotinn  like  that  in.  a  fn  e  \orte.v. 
Its  \ehicity  |iaiailel  ti  >  the  axis  n{  the  stream  is  L^reater  .it  the 
inner  than  ,it  tlie  outer  side  'it  the  curve. 

Thus,  to",  the  \sat(i'  iti  ,i  M\ir-banl-,  Hows  more  quiikly 
aloni,^    courses    adjacent    to   the   inner   b.ink   ot    the    bend   than 


|::-1 


aloiiL,'  courses  adjacent  to  the  outi  r  'I'he  water,  in  \'irtue  of 
centiitu;^al  toi\a-.  press.  ^  nutu.irds  s, ,  th.it  tlu'  w  .iter-suitace  of 
;i  trans\erse  sri  tion  .Viv,.  I  501  has  aslope  rising'  upw.ards  trum 
the  inner  t<i  the  outer  bank.  Hence  tlie  free  le\el,  t'ir  any 
])article  of  the  water  ne.ir  the  outei  bank,  is  JiiLjher  th.m  the 
free  levil  tor  an\'  p.irtK  le  in  the  same  trans\erse  section  ne.ir 
the  inner  bank,  but  the  teiidenc)'  to  tlow  from  the  hii_;hei-  to 
tlie  lower  level  is  counteracted  by  ■  entrituL^al  action.  Now 
the  w.iter  imniecliatcly  in  contact  \uth  the  bottom  and  sides  of 
the  course  is  retarded,  .md  its  centrifui'.d  force  is  not  sufficient 


t'-! 


KIWKR-BHNDS.  27' 

I'l  h.ilaiiL'c  tin:  r)rc'^-;uic  due  to  t!ic  ^rci'  -r  iK-jitli  at  the  mit^iilc 
dftli^'  iniul.  1  hi^  water  therefore-  tends  to  tlow  tVoiii  tlic  outer 
bank   to\vard.->   tlie   inner    ^I'ig.   15 1;,  carr\in;;    witli   it   detritus 


Fig.   151. 

N\liich  is  (!e])osite(l  at  the  inner  hank.  Simiiltaneou-h'  with 
tile  tlow  of  water  inward--,  the  mass  of  the  water  must  ncccs- 
sarii\    ilow  iiutwarils  to  take  it-  ])Lu-e. 

13.  Flow  of  Water  in  Open  Channels  of  Varying  Cross- 
section  and  Slope. 

.Issinuf'fitins. — ia')  Th.it  the  motion  is  stead\-. 

Thus  tlie  nie.in  veh)city  is  con^t.int  for  any  f,'ivcn  cross- 
section,  but  varies  i^riuiiia/ty  from  section  to  section. 

(^/')    That  the  chanL,re  of  cross-section  is  also  -gradual. 

ic)  That,  as  in  e.ises  n^  /(iiihTiii  motion,  the  work  absorheil 
b\'  the  frictional  resi-t.im a-  ..f  the  i.h.innel  bed  .m'\  sides  is  th> 
onl\-  iii/iiiKiI  work  wliuh  need  lie  taken  into  consitleration. 

Let  I'"i,L;.  i;j  npre-iiit  ,1  lonL^itudinal  section  of  tin- stream. 
1  he  iTuid  molecule--  which  arefuind  in  any  plane  section  dl' 
at  the  commeiKemeni  of  an  interval  will  be  found  in  .1  curved 
-urftice  r/i  .it  the  end  of  the  interval,  on  account  of  the  diltereiit 
velocities  of  the  flui'l  filaments. 


p'"i 


2-2  (H-IWhIS    Oh    y.-1KYI\C,    CROSS-SECTION. 

Sui)i)ii>e  that  tlu-  ni.i-^s  ofwatrr  hiuindcd  1)\'  the  t\Mi  trans- 
\ersc  section'-  ill'.  <7  c unc--  int'i  th<-  positmn  iii/i^i;  in  a  unit  n\ 
time.  Tlu'ii  the  ihaiv.je  cif'  imtie  eiuTL,')-  in  this  mas-,  i-  (ijual 
tc  ihe  aliJcbr.Jc  -uni  ot  the  we.ik  dnne  t)\   -ravit}-,  of  the  uork 


done  1)>  iin"-surc  am'  "f  the  work  done  ai;ainst  tlie  frit  tional 
rosistante 

(  haiii^c  of  Kiiittii  Emtf^y. —  This  is  evidently  tlic  difference 
between  the  kinetic  ener^;ies  of  tlie  masses  ,t\s^li  and  •/',,/, 
since,  as  tile  motion  i--  steady,  the  kinetic  energy  <if  tlie  mass 
between  <v/and  </  lem.iins  constant. 

Let  .-J,  be  the  area  ol  the  cr<iss-scction  ah. 

•  ■      ;/     ••     ••    mean  velocity  across  this  section. 

•  ■      7 \elocity   at   this   section  of  any  ^iven    fluid 

til.iment  of  sectional  are.i  a. 
Let  .-   -     «,  ±   /'. 
Then 

,7,//,  --  it.f.-i      and      i"(<il')=0. 

The  kinetic  ener^fy  of  the  mass  ,;/■,,/ 

=  "'i-("^'»)=  '''i^/iw,  ±   \"f\ 

=  ^  i- ; «^«,»  -b  3«,' ' '  +  3//,  ( •••  ±  / '') ! 

3A' 


smcr 


2:wi/')-0     and      ,V/,  ±  /'  -  2«j +7', 


m 


■■!t 


CH.-ISSII.S   Oh    l-.tRYlSCj    i.KOSS-slCriOS.  -li 

Now    27/.  -f-  :•    i-    c-vi.KMilly    i...-,itivc.       ll.tKc    the    kiiuiic 
fiuT'j^')'  of  the  mass  a/nd 


'* 


n  l.cin-  a  .ocffRUi't  of  o.rn-ai..n  uho.c  v.ihir  .Irp.Mi.K  upon 
tlu-  lau  of  tlu-  di>tnl..>ti..n  .  .f  tlu'  \>'locity  ihioiioliout  the 
section  ,//'.  It  1^  poMtivc  aii.l  -rcitcr  than  unity.  .\^^:n,„- 
tlMt  <r  ha^  tiic  >.imc  value  fnr  the  sections  ol-  and  ./.  1  Inn  if 
./  ,  /,,  an-  tlK-  area  and  ine.m  vel...  it>  ..t  tlu:  t..in-.ve..e  .eUion 
</,  the  kinetic  elier-N'  of  the  mass  <;/>/( 

,.'■'././/.'. 

Hence    tlie   change   "f  kinetic   ener-;.    in    the    ma>s   uiuier 
consideration 

zvQ  h;  -  /',' 


i 


since 


./..«.=  <J  —  -'l"! 


H;W-  ./,'».■  /'.)■  f;;7».-//r.— Consider   .in>    tluid   filament    mn, 
the  deptli  of  III  below  the  surface  hein-^  .r,  .  and  of  //.  y.. 
Let  J  be  the  fall  in  the  <urface-level  from  ,/  to  , . 
Then  the  fall  from  m  to  ;/ 

■.».-;>• 

.uid   the  uotk   done   by  -ravity  ..n   the   elementary  volume  r/Q 
m  a  unit  of  time 


'!sl   , 


274 


<:H.-1SMl.S    Oh    r  IK)l\i,    (A'O.nv  s/v  Tl()\. 


Hi 


ll'crl  liiiiii  hy  rnsiiiri-. 

'I'lu.-  prcssiifi'  ])rr  unit  uf  ana  at  in  ~  TiT,  -(-  /",, ; 

"  =  '^y,  +  A,; 

/,,  hiiiiL;  tliL-  atiiio-^phci  ii   inosurc. 

Jliiuc  the  work  (liii-  t..  tlu'--r  ]irr-,siMT^  ]Kr  unit  Mt'tiinc- 

^^  r/O-'T,  +  A.I  -  'iQ\'<.'y.  -t  /„) 
=  rt' .  '/(j(  j'l  —.I.  I. 

']  lui^  thf  /<'/(//  WMfk  (liiiic  tiy  i;ia\it\    ami  l)V  pro-^siirc 

^  i-  t.'  .  ,/C>  .  -I  =  r.'C'r 

fiir  tlic  in.iss  iiihIct  i unsiiicr.itiun. 

/('('/■/•  ahioibcd  by  /r/i//,!//.  -  I'nw^'tdvr  a  thin  lanini.i  '>f 
water  <>i  tliii  kncss  (A,  linuiiiKil  hytlu  tra!i'\iT-i'  planes  .r.i . 
yy.  tlic  ilistanco  of  .j.i-  from  ir/>  htin^  .v. 

Since  the  clian},'e  of  velocity  is  ^fratlual,  llu  naan  \floiity 
fr<ini  A. I  to  I'l'  niay  be  assumed  to  lie  constant. 

I.t  t  //  he  tlii--  mean  velocit)'. 

■    /'  he  tile  wetted  perimeter  at  tile  sietion  xx. 

•    ./  ln'  tlie  a.ea  of  the  waterwa}'  at  the  ■-ection  .r.r. 

Then  the  work  absorbed  in  friction   per  second  from  xx  to 

and  the  tcit.d  \\ork  ab-oihed  Ixtweeii  ,i/'  and  t f 


i  bcin{j  the  tlistance  b<tw  een  iil>  .iiul  i  /       I  icnct 


LHA.wr.lS    (.)/     yAKMMj    CROSS-SECTION. 

'   r  J\u) 


nnd  ttuTifcirr     .:       m '-  -i_ 


'.is. 


^X 


„  /-'(in  ir  ,      ./  .... 

Tukc  ^/—      and       ,,  =  w.       1  lull 


It:  —  II,'        /    /  "■  / 


IM 


If  the  t\\"  ])lanfi  (f/'  and  (/arc  iiuU-ruiitily  m-ar  niic  aiiotlici 
(Iij;.    15.1,.  tile  last  (.-(luation  c\ulciitl_\-  ^ivcs 


tt  /     ir 

t/s  z^      II  .  tin  4-  (A. 


<2) 


uhicli  i<  tlu'  fiindaiiuntai  diffcriMitial  ciiuatioii  of  steady  varied 

Kiiitmn,  (/.:  heiii^'  the  (all  i)f  -iirf.iie  lc\el  i"         ^^        ^^■ 

tlic  distaiKf  ^A'.  ^ — ■ 

111  tlic  fi;^'iiri-  ixii  is  di.iun  ]iaralkl  ti>  tlit;         ,».^  / 
htd   and    ,iii      !->    liori/niUal.       I'lie  distaiu< 
a  ,■  ma\ ,  uitlidiit  sL-nsihlc  I  i>.  .r    In- assuiiui 
tcjiial  to  liz. 

Also,  ti   ii'        I     (III         I     'A.  very  nearlv. 

Hence 

/,A  --  an'     -  lit  -\   It  t        till  -\  li^.  .  .      131 

Substitiitinj4   the   value  <if  </-   from   this   t<|ii.itioi)   in   ei|iia- 
tion  (21, 


IlG.    153. 


/  .  lis  —  till  =//.(/«  4-  ■     —  .  (A. 


Also,  since  All        (J,  .1  lonstant. 


(4) 


o. 


and  it.l   ~  x    till    ver>  nearly,  il  i  is  the  width  of  the  stream 


!   il? 

m^ 

1 

\'^  < 

w 


tjff  ai.-IWhIS   ()/■    i:-lK'il.\(,    I  KOSS-S/i.TION. 

TIicrcri>|-.,' 

.  li/u  -I-  i(.\  .  ii'/i  —  o. 

.mil  liotKo.  !))•  (_<jii;itii)n  (4), 

.  ..  ir  .'•        ,.  /'  ir 

I  .  <l<  —  ,1/1  —  —  tt  lili  A-  '  d^. 


TlliTct 

1  lie 

f    rr 

/    u^ 

ds  "' 

1  — 

»'  ^.C 

ir  X 
I  —  *i-    . 

111  2i,'-/ 
irx 

(5) 


I.'jt  tilt-  |)r>^iiii)i\  nfjiii)-  I'.iiint  ,1  in  tin-  Mirtaci-  he  defined  ^ly 

its   pcr|H.-iidicul.ir  di^laiue  //  fnnii  the  l)rd  .iiid    l)\-  the  distiiue 

,»   of  the  triiiis\cr'^e   soctiun    it    <?   tVnm   ,111    (iri;^nii    m   the   l^d. 

^,        dli 

Then  -7    is  tlie  t.m;,'ent  nf   the   .nude  whu  li  the  tangent  d.  t!ie 

its  ^ 

surface  at'?  iii.iUe>  w  ith    thi:    bed.       It    i-.   |.Msiti\e   m-   iieLjativc 
.iccorditij^  .IS  the  depth  iiiercascs  or  diiiiiinshe--  in  the  directinn 
i)f  flow,  thus  dftiniii;^  two  states  ofsi.  ,i(l\    varied  iiKitMii. 
Hetwcfn  tht"~e  there  is  an  intermedi.ite  st.ite  detlned  by 

'///  /    //•' 

=  0  =  /  — 

lis  in  2:' 


p:^ 


III 


IS    tile    e(lii,iti"n     Inr    -te.idy    tV  •w     \Mtii    iinifi'ii. 


ninticitl 

1,1't  / ',  .1/.  //  111'  tlie  eorrespiindin^^  v. due-  <A'  u.  in.  li  in  tin 
vase  of  uniform  niotmn.      TIilii 

/  r 


anil  •^^\.   I  5)  lu-e. lines 
dl, 


'   ^  .1/2.4-         ''-'''■ 


.1/  ,C 
m  (' 

/('    X 

\ 


HI 


I  —  11 


t  —  «- 


.^) 


'/■; 


c//.-/\\//.s  "/■  r.-iKYis'O  f;w').s>-5/;(.r/('.v. 


277 


If  the  ^celuiii  lit  ihc  tliamicl  i-^  a  rci.t.Lni;lc 

a// 


.7/ 


A=-.i/i,   xhu   ^xin\    />i=^_   ,     ,/^.     nul   .l/_^_^^^^. 


Suh-^titiitinL;  tlH-<c  vaiur^  m  en.  i7'. 

.1+2//  ,7/,' 


,///         .'       J-  +  2// 

lis  tr 

I  -  n- 


Vii 


.  .  •  («^ 


Tlircc  cases  will  Ik-  m  .iiM.lcre.'  ..ul,  m  each  i.i-e.  a  line 
/•(),  drawn  paiall.l  tn  the  bed.  represent-  the  -urtaLe  ..t 
inutoy,,,  iin'tiri,.  U  hem-  the  distance  between  PO  ■>"^'  l'"-  ''^•''■ 
(Asl.  1  ^x!r  ■  .;■/'  '""l  ''''  '^'  ''■  '■'.-;  "'-^ 
''''''  ,,  poMtive.  and  Iherefi.re  //  nureaM-  m  the  dnt.tioii  ,.1 
now.  Thus  the  aaual  -nrlaee  J/.V  ..I  the  .tiean.  i-  a  hull) 
above  the  hne  PQ. 


"* 


Fk..    154- 


Proceeding'   np-stieain,  //  becomes   more   and  mm 


(      IK  aTlv 


f(iual  to  //,  .so  thai  the  numerator  of  eq.  (8),  and  tiuntwi.   al  o 
11,111, ixiniate--  more  and  nioie  tlosel.v  to  :tro. 


'1 


\'^Am,  proceeding;  down-strcam,  //  increases  and  tt 
.limmishes.  so  that  both  the  numerator  and  denominator  m 
cu.  (8)  approxmiate  nior.    and  more  closely  to  tiic  vahii-  umtv. 


.ill 


and   therefore'       become^;  more  and   more  nearl\  equal   to  i. 


its 


the  -h.iie  icrrespondin^  to  unitorm  motion. 


!i 

i 
1 
j 

a;! 

278              (:h.-i\'\i;!s  oi-  i'.-ii<Yt\(,  cko^s  sucrios. 

|i 

Ill'llLi-     Up-sttr.iin    J/.\'    !•>    .IS_\  lll])lMtu      Im     /'(J.     ,111(1    (ioUIl- 

ii 

strcain  MX  i--   ,i-.\  iiijitntu    to  ,1    lnn  i/nntal  !nu'.       I'lii-.  tnrm  of 

1  il 

surface   i-    iiiodnt  nl  ulicn    a  weir   is    Innlt   acros-,  a   '.liaiiiu-l    iti 

1 

wliicli  till    w.iti-i   li.iil  ])revi(iusly  flowcil  w  illi  a   umrorni  miitJMii, 

1 

(■  \M    II        ,>/,■  ^.-//      aii.l      //  ,■  h.   1':-     1  ;_;. 

11 

I--  now    iieL;ati\e,  aiul  the  ileull;  iiiiiiini>lie>    in  the  ('.inx - 

(IS 

tion  I  if  lliiw 

I   ]i---ti(Min  /V    iticreasi-s    .uid  a])|)i 'Mt  Ik--  //in  value,   »>  tll.it 
MX  is  ,i>\  niiitdtii    t..  rO. 

1  )(iun-->tre.uu  //    (liiiuiii^iie<,   ,7  iturea--es.  .uid  tllerefure    the 

\a!ue  of      ,    i--  ninre  and  innri    m-.uK    luu.d  tn  liiiity. 

Thus,    HI    the    hunt,    the    deii.  mun.itnr   in    ^:^\.    I  ,S )    l»eeoines 
'/// 


/iTii.  .uiil  tlu  letore 


,/v 


U  in  e     the'  11  \     ut(Me.lti-s    tJMt   ,lt 


a  certain  |iMiiit  dow  n--tre  nil  t!ir -uil.u c  line  .I/.\  t.ikes  ,1  direc- 
tion 'Alikh  In  ,ii  tiL;ht  aiiLjles  to  the  i^cner.d  direction  of  tlou  , 
T'ln'~  is  t  (intrar\-  to  the  fundament. d  hNjiothesi-.  tliat  tile  tlui<i 
'il.inient-    llou   in    ■>ensii)ly    pai.dlel    line-.       In    fact,    before   tlic 


Uli- 


■J 


ill  |:-< 


limit  Could  he  leaihid  tin-  h)  potilciis  uould  ce,i-.e  to  be  even 
approxiinateK  tnu.  and  the  general  iMjiMtion  would  cease  to 
be  ,ipiilu,d)le  i'hi--  lorin  of  w  .iter-surf  ice  i>  prodnced  w  hen 
there  IS  an  .il)ru|>t  depres^inn  m  the  bed  of  the  -^tre.im. 

l'"i^.  156  shows  one  of  the  .ibrupt  fills  in  the  <iaii}^es  cui.il 
as  at  first  constructed  I  he  surface  of  tiie  uater  How  inj^  freely 
over  the  crest  of  tlte  fall  took  a  form  -similar  to  .I/.\'  Ix  L.w  the 
line  PQ  of  uniform    motion         Tlu-  <liminittioii    of  depth    m  the 


C7/./,V.V/:7.s    ()/-    r.-iKYlSC.    (.ROSS  SfCriOS.  ^79 

approach  IM  thr  tali  ..m.c'l  an  wKrca^.-  u,  tlu-  x.lnuty  ..fflow. 
vuth  thorcvaU  that  Mr  ..^cra!  nul.-  atun  r  the  tall  a  scnuu. 
ornMon  m-  the  bc.i  an,i  .ulc.  tnnk  place.  In  ..nicr  to  rcnu.ly 
lln^    tcnipnrarv    ^ven'.  we,v  c.>n~.trucrcd  m,  a-^   tn  rai^c  -he  level 


.v 


Fic.   156- 

,,rthe  water  until  the  Mirfao    hn.    a.MinuMl  a  f.-rni  ,J/.V  e-.rre- 
spnnduv.-  ap,.rnx..natelv  t.,  /U}       In  ...n.e  .a^e.  the  uater  uas 
,aiM-,l  ahnve  it^  nunual   hei-lu  and  a  luKkuater  pmdiue.l. 
r\-l.   111.       air    .>  ,1^/1      .iil'l       //■"■■    l-'i'-^      ^'-7- 

'^''    i.  ne'^atue  and  the  ^urlaee  Inie  of  the    stream  1,  uhoUy 
(A 

ahM\c   rQ. 


•*- 


Kr.;.  157- 


,/// 


r.  ixiniates 


If // ^n.uiiially  iiKiea-es.  n  .lruun>he-  an.l    ^^    appr- 

tn    -  /  in  value 

If  h   K'tailually  dnmni^he-,  it   a[.proMiuate.   to  //  m  value, 

and  in  tlie  hunt     .     =  O. 


11 


28o 


cH.-is'N/i.s  Oh  i^'.^iKYixc  <:ross-s/:cti()m. 


lu'twccii  tlic-M.'  two  cxtreincN  tluTf  is  a  \.i1iil  i<]  /i  lur  which 
the  (Icm  iiiiinatDr  of  ctj.   (>S )  bccdnics  lul,  vi/., 

/ 

,     ,  ,■  '''/' 

and  tile  c  iincspoiKimi;  value  c,t  i>  intiiutv. 

i/.'i 

I  lui^  iiiR-    part  (it   tin-  ■-urfacc  linr  i-  a--\m]nc)tic  tn  /'(J.  the 

hue   (it    iiMiturni  mnticiii.  aiicitlicr   p.iit  i-   asyniptntit   tn   a   hori- 

/iiiital  hnr,   while  ,it  a  certain  jxiiiit  at  whicli  the  ikpth  i^ 

/<■  —  (*     , 


the  surtace  nl'the  -tream  i-  normal  t'l  the  hed- 

'1  llis  is  eiiiitrary  tn  the  hiiuianieiital  h\  pntln-sis  that  the' 
nui<l  tilanuiit--  flow  ill  --eii^ihly  p.iiallel  Inn--,  and  the  general 
e(|iiatinn  nil  icin;^er  repri-'sents  the  true  ei'inliTinn  nf  llnw. 

In  ea^t-s  siali  as  this  there  has    lieen    an    abrupt  ri'-e    (if  the- 
surtaee  dt   the    --triMin,  and  what  is  called  a  ■ ' -taiidin^;  \\a\e 
iia--  been   pri  niui.  id. 

In  a  >tr';a!ii  nt    dejith   //flnwirv^  witli  a  nnifMrm  \ehieit}-   ^\ 


wllieli 


'■   y  „ 


1.  I  'ii-.tr  lie t  a  w  eir  si  i  a--  t'  i  im  rea'-e  the  <lepth 


to  //, ,  \\  hit.  Ii  is 


<rr- 


P'.:i 


'1  lu'n  in  '>?)e  piittHin   nt    the   -tiealli   ne,u     the  wiir  the  depth 
IS  >  ,    wliiK-   turther    up   the    ^'re.iin    tiie    dei>tli    is  <     -      . 


S1\-1M>IS<,    lV.4yF.. 


281 


f-i 

'llni~  .it  ■-"me  intrrim-cli.itc  pMiiit  the  depth   --ft-    ^  .  the  I'lm- 

."> 

,//'/  ,  ,  •  ■  J 

^n.  iinhiv^  \,ihic  of         Ih-iiil;    v:  .    and    at  this    piHiit    a    standiU'^ 
'  "^  (/■■. 

7<  i/,"i    !S  pycilliiid. 


N  I  i\v 


/r- 


.)//  =  ///, 


and  sincf      //  <  « 


r'^ 


r\.vn\  thtrcfiiro 


<  >x        I, 
■2g  g 


2(X 


wliicli  condition  must  be  fulfilled  for  a  standint^'  wave. 
Ha/.in  [,Mves  the  follouinL;  table  of  \alues  (.1  /: 


N..liirf  ..f    Bf.l, 


/  ^'  \  St.iiuliii^  W.iVf  1'ru.luced 


Very  miumiiI)  1  niirnird  M.irfacc. .  . 

Ashlar  or  lnn  kwmk 

Riibli'.e  m.isoiiry 

Earth 


Sl„, 

bt-tnw  vvIiilIi  slaiul 

lilt;  w..  vf  IS  III! 

(.i)ssil)lc      In 

Mftrcs  per  Metre 


.00147 
.00186 


S1<.1«-  ;n  Metres      l,,,^,  „^p,|, 
jK^r  Metre  I. .r  ,„  Metres 

Feet  \XT  boot) 


.1)0231; 


.01.275 


£   .i>02 

•!  .003 
'  .01)4 
i  .003 
-.  004 
f  .006 
i  004 

-,  .(lOO 
(    ..)IO 

(  .006 
.010 

( .015 


.oS 
.1)3 
.02 
.  12 
.oA 
.03 
.  V) 
.10 

.08 

.1)6 

.47 
.28 


A    -t.mdini,'    wave    r.iiely   oiiiirs   in   Ji.tnniU    with    r.trtht  11 
bed^,  a-  tin  ir  -lc.,)e  1-  .iluK.-t  .d\\.i\  -  le^-  th.m  the  hunt,  .'"Ojj 


\ 


f 


I 

'i  . 


Tin-   formation   nf  a   -t.mdins^  wave    was   fir<t    oh-i-rvcd   l)y 
I',ii!i>iK  in  a  -mall  nia-onry  canal  nf  rcctan-ul.ir  ^cctiuii. 
The  width  Mtthc  canal  =  0"'.32  5  ^  x  ; 

'•     sl<>i)c  (=    ')  of  the  canal   ^-        .023; 

••     unitorni  \clni.  ity  of  How  \"\>u)     =   (  ; 

•'     ilcpth  CDiTi'-pondin-  to  ('    -  o".o64   --:   //. 
A  weir   built   aero-.-,   the   canal  increased    the   .iepth    "f  the 
water  ni.ir  tlir  weir  tn  o'-'.jS;         //,. 

It  ua-  found  that  the  ■•  uniform  n'-ime  ' '  w  .1-  niaintameil 
up  to  a  pomt  within  .i'"..:;  ot  the  weir.  At  thi,  j.oint  the  <lei)cli 
-uddenly  increa-ed  troni  o'\oIk\  to  about  o'Mjo,  and  between 
the  point  and  the  weir  the  surface  of  the  stream  was  sli-htly 
convex  ill  t'  irni  1  !■  il;     i  ^S  i. 


With    the    precedm-    data    and    takin-     a  11, 

and  i-  thcretore   "-    I   at  a  section  (?/',   i'i.i,',    1  ;'>. 
At  the  section  < ,/, 

//  .004 

«=  ^^  r=  \^  X  I. r.9=-o"v 377. 


5 


if- 


.o^  ;  and  i-'  therelore    <  I  . 


'I'lui-    the    expression    i  is    ne-ative    tor    a   section    a/) 

^  ','        .    and    'xisiuve   tor   a  section  (./,  >i) 

a  h —  : 


that  I  must  chan;^e    ^u;n  lu'tueen 


,///       ,,     , 
those   section-,  ,ind  wdl    then 

bicome  intinite 
Kir.,   i5<).         "  Ciuisidcr  a  portion  "ifa  stream 

bounded  in-  two   transverse  sections,  ,il>.  < v/,  in  which  a  >taiid- 
inij  wa\  :  occurs,   iii;     i;'^ 

Assunu'    that  the  tluid  filament-    l1ou    ,100--  the   -eiti  )n-^  in 
sensibK-  i)arallel  lines. 


li 


I 


sr.iM)!\<,  n.'iyf:. 


'■^5 


Let  the    \"cl"citie--    and  .ux'.i    at  scetiini    iii>  i>e    (li'-tiiiL;iii>lR(l 
1)\'  the  --uftis'    1,    and  th'.-e  at  -r/li\    t'.u-    aittix   j  1  lien 

(."hanLTc    <if    niiinientiini    ni    di-  » 


leeticMi   "t  lldw 
1  leiKe 


) 


inipuNe  in  ■^.lnlc  directinii. 


;  >;7</rOr'.,  —  i'r^jf,)'"! '  ^  '^'^-^Jy  -  •'.•.''.>■ 


nd  thereloie 


i^iar.r  —  ~ir:\-<  =  ./,  r,  —  ./,)■ 


.      (9) 


r  .    ;     l)eint,'^   the   depth-;   hel.iw    t!)c   -;urfaee    nl    the   eentre-^   of 
}4ravit\   '.f  the  sections  ,t/>.  id.  respectivel\-. 

N..\\  ,   fj  _z;  7<|  ■\-   \\.        1  lieretnr( 

Also,  as  ahc.iil)    shoun, 
and.  nei^deetini;    / ',  as  eom])ared  with   \n^. 


Thus 


and  hence 


•  'i"r' 


3 


4-  21  =^  '•''.?,«,', 


Of    +    -  ,    .  -r 

where  rt   =  .  and  is  1.03.^  \t  (n  =   1. 1. 


i    ' 


-'^4  >/.-/,%/ /.'A  O    H.IIL. 

SiuiilarU-  it  ni.iy  b''  --liduii  that 


A' 


(/J,///---;,//;)  -:  ./,.!•,  -  Jjv 


(  I  O) 


Let    the    scctmn  i  if  the  i  .mal  he  a   rectaiij^Me  of  depth  //j  .it 
ah  ami  //,  at  <</.       Then 


,/,//,  =  «,//,; 


//, 


//, 


'rileietiii'e,   hy  eciuatnai  i  lOi, 
.III, 


uliicli  rechice--  to 


<i  II,' 


2//, 


//:■  -  ff:\. 


-u.. 


//   =  //|  satisfies  the  <c|iiatioii    and  eorresjiotids   to  a  c<-)n(lition 
of  uniform  motion. 


Al>o, 


a'u,'  _  If,  //„  +  //, 


.      (ii> 


In  l<i(li:iies  eanal.  «,  ---  i'".6o,  //,  —  0"'.o64.  Substitiitin;^ 
tilese  vahies  in  ((luation  i  M  ,  tlu-  \al'ae  ot  //,  is  found  to  he 
O'"  ii'.  which  a.L;rees  -.oniewii.it  ehisely  with  tin  aitiial  mcas- 
urenunls. 

N.H  — The  cot-ffii.  unts  a  .\.'A^\  m  ha\e  not  been  \'ery 
acciiratel_\'  tietermine,!,  hut  tlieir  e\.u  t  xahii-  are  not  f  ;-;rcat 
nnpoitanie,        1  he\   are  oftMi  taken  eiiual  to  ii>iit\. 


A7  ///.vr  v.vvs  /  /;r 


2S5 


14.   Longitudinal  Profile  and  Ruhlmann's  Law.      In  the 

prccc-diiii'  ,irtii.lc,  nut  /■  1/  —  /'-  I  ^   i        ,,         , 


It  :hc  tran^Vfr-^i-  iiinfili-  I1.1-.  hrcr,  ■IftiTmiiu-d,  the  \,ihic  nf/-' 
corrc-^l)iiiuliii;4  in  tlu-  depth  //  .it  .ui\' [iniiit  •'  i..iii  hu  .it  once 
toiiiid  .uid,  hy  ir.'Mii-'.  "I  tlu-  Li-^t  ciju.itK  m,  the  Mirfaee  jirntllc 
between  the  depth-.  /'  .md   // e.iii  he  e,i^il\-  ph^tted 

Let  /■,,  /,.  /,,  .  .  .  '.jc  tile  \',ihie->  "t  /  ,it  .1  serie-.  of 
poults  ,lt  whieh  the  dejith--.  (h'll'erini;  ^ut,  ce^^iV  el\-  h\'  a  -.111, ill 
quantit)-  ,/'/'/,  ;iie  /',  ,  ;},,  /),.  .   .   .  resjicetu  eiy.       I'lieii 


./.v,  =r  /•,  .  ,///;       ,A .  ^  /• .  .  ,///  ;       ,A ,  =  /•■.  .  ,///  ; 


nnd  the  ei  >n  .'^p.  iiidiiiL;  di^t.itKes  .v,  ,  S.,,  S ot  t]le>c_'  points 

from  I  '   ire 


i/s,\i/s.,  </s.-\-i/.\, 


>    J . 


^^,/V:-.^, 


I'.XAMl'l  I  .  A  (.lit  ot'  reet.uv^iil.ir  section,  uitii  .1  f.ill  of 
I  in  lo.fxxj,  i-^  10  ft,  wide  and  deh\<-r>  4''  in.  It,  of  water  ]ier 
second,  ,\t  a  Lcrtain  point  thi-  depth  is  increased  to  (  ft.  In- ,1 
daiii  .\  .^unii;i;j  that  tlie  lace-.  ol  the  cut  ,ue  not  \  ei  y  sinootli 
and  tli.U,  t  onse([uc-ntl\  .  .noiu  illa\  he  taken  a-,  .111  approximate 
v.ilue  ot  /'.   then  tlie  deplli,   //,   tor  uniform  motion  i:.  ;^i\cil  lj\" 


(,;;^r='--"; 


10// 
104-  2//' 


or 


So  4-    {!,//=    5// 


jind  .III    ij>pro\imate  solution  '.f  thi^  iiiu.ition  1^  //   ^    2.9  ft. 

Ihe  following  I'ahle  van  now  lie  c.isily  prc().irctl  for  a  series 
nt  depths,  comnuiu  in;,;  .it  tlu-  d.iin  .uid  diminislnn^f  successively 
b\'  ;  ins, ,  ,»  beuiL;  unit\- : 


f: 


286 


A7  //A.Vf,-/,N'.V,s    /,,/«-'. 


P 


19 


t:.i 


4.00 
1  7S 

1  •$ 

(.(JO 


/4 

40 

37. s 

3S 

»'-5 

10 

iS 
'7.S 
>7 
16,5 


10 


■  I 


i! 


*■« 


991406 

.00045 

^  JO  ^  I 

9 

550.4 

1 

U91943 

..>j«>4l      , 

J4l'(t 

7 

C1048 

'       < 

7;H 

)J148o 

.UUUUJ7J 

ifc'ri, 

'' 

6(170 

>>• 

"17 

'W3017 

■  ««»3JS 

'J'^4^ 

< 

741. > 

••1.177 

•i»3554 

.OlJOitj 

.(l'*4 

7 

Ii54<> 

»7. 

■55 

The  tcntli  coliiiTUi  l;im-s  tlio  dist.iiin  >  Ikiih  liic  il.mi  nf  tl^.- 
sections  in  uliicli  the  depths  are  ^75,  3.50.  ^.2$,  and  3  ft 

l\iili!m. inn's  I'lrnnil.i  ■  tlie  distance-  between  two  sections 
between  winch  the  lieptii  nf  thr  \\ater  ijraduaily  increases  t'mni 
j'  +  // t.)  )■-}-//  is 


7  s    /)^        Jy], 


tile  I'liiu  tmn  /■('     '  luin^  },'ivcn  l>>  tin   tojluwin^  table: 


4/7) 


'(^) 


O.OI 

0.0s 

O...J 
0.04 

It  of> 
11.07 

«..o8 
o.rx) 
i>.  ii» 
•  1.20 


o.<io<>7 

'    0.3 

1.  MJ^ 

"2444 

<>  4 

1 .  5ii() 

o.3S<>j 

0.5 

1  .(<(>■  I 

0  4889 

0.() 

S .  7.)^.) 

0.5701 

!          0.7 

l.qs<>0 

0.6376 

0.8 

2.<>4<)5 

O.fHJSS 

0.9 

2.1fi«3 

0.7472 

1.0 

2.2H3., 

o-7<>3.' 

I.I 

2.3'»7i 

0.8^53 

1.2 

2    <'.8l 

1     Mfi  1 

1     ^ 

:  '  1-  , 

a) 


//. 


14 

2.7264 

1.  K 

2.8337 

1.6 

2  ()40l 

'■7 

3.045H 

1   •< 

3-i5<>i* 

1    q 

3  3553 

2.l> 

J-ii'H 

2S 

.3**745 

3.0 

4-.1»43 

^n 

4.8010 

I  '• 

;    I')'' 

.\pl)l>inj,'  tliis  fiirinula  V>  the  prei  edinj;  e.\.nnple.  in  ord.r 
til  lietermine  the  tiistancc  between  tl>c  3-  ami  4-ft.  depths,  at 
tin  dam 

y     1.1 


CH/1SML   ('/■'  Rl-CTASCri  .-IR  SF(:riO\    .1\li  SMAll-    sUV'i;.   :>^7 
arui.   li\    intcrpiil.iticiii, 

At  tin-  ^-I't.  (lt;])th 


7?  =  2.0  =  •°-^'*'^'^' 


aiul 


lIlIK 


.(;;)  :.4P3. 


^  =    ^^,^^  -^'CV  -    4323)  =  30.293  rcLt. 

.000  I 

15.  Channel  of  Rectangular  Section  with  a  nearly  Hori- 
zontal Bed.  In  tlii-;  ca-^r  /  i-<  vtT\-  -^niall  .ind  nKi\-  Ir-  disrc- 
Ljanl'  1  ill  t-ii.  (4),  Art  1  >,  uliii.h  ni,i\  tlu  rcfurc  hf  u  rittcii  in 
the  form 

-.•:  >>'  J,       -.<  "' ''" 

lis  =  —  -    .     Jli  —  »  , 

WwKxItu-   (_J  =  aconstaiit,  ami  thcnti.ic //  .  7// -! -'/    .///       n 
Also. 

xh 


m 


I  -\-  2/1 


Hence 


lis   —    —   i,v>.7      .^./.x  -f 


.ill 


^QKx  -^  2I1)  ^  Av-''  +  2A" 


IntcyratiiiK, 


1-7/ 


r  beini^  a  constant  of  inte^jration. 


s.'^s  ( //.w.v/-/.  Oh-  r.RE.ii   uinii'    is  <:oMf'-thvi>  uith  nirin. 


t:::4 


Hciu-c   tlu    distaiu--    V,        .,  lu-twcrn    two   point.   ,it   w  iiK  !i 
the  .Irptlis  ;irc  //,  ami  //,  (  -    //,)  i^  .i^ivon  l.y 


l,Q-  /,     ,.  +  2//  ''    I-kJ,,     •' 


/ 1: 
I     2// 


TIk  l.i.t  term  i"  iisuallx  vcr\  >m  il 


aii'l  iiia\  Ik-  di^rc^Mnicd 


uithmit  appixH-iahU-  iiror.  aii'l  tlu-r.t.  .re 

.,'  /*•    /,\:^- 

^  -  Ms.     -  v.i    '       '    ■    '''' 

I  -    • 

a  formula  t,v  mran^  of  uhiJ.  tlu-  .U.^Iku-.    "'^'V  l'*"  '""'"1 

i6.  Channel   of   Great    Width    as    compared    with    the 
Depth.      I'l  this  c;'sc 

A  =  .r//     .iihI      /'        1  ■  api.io\imat<-ly. 

1  lu-nior. 

„/  =  //      aiu!      .'/       // 

Also. 

//■    .         //•' 

!k-iu.-    .-'i    '7K  Alt     I  j,  ma>  1»-  uritim  m  tlu-  form 


ai 


',l/i  ~   !  It 


'''+2'  _   ''      I    liciiu'  th.-   risi-   or   fall   above   or 
below  the  Mirtacf  of  uniform  motion         I  I"  "  ■/''        H    •'  -  •'"-' 


laki 


// 


'•K 


Inti-i^nitiiiLS 


,   bciiv^  a  constant  of  itit<.-t;ration. 
This  cquat.i.n  m  i\-  be  written 


I         I 


2.V 
// 


(i  - "-_);0(->  +  '^ 


(2) 


iivl  iRtuc.ti  any  two  points  ^^  ami  jj 


h' 


f,  _ '").^.(.-, -0(:r');,  .    (3) 


r        chanf,'o  in  depth 
ttic  ar^;un>ent  hein^'  ^^  =    ..ri^.jnal  depth 

In  tln.asrofa  dam  built  across  a  channel  in  uhich  the 
water  had  p.cvionsly  flowed  with  a  uniform  motion,  Case  1. 
Art.   13,  in  the  limit. 

si  =  A  -  -//  '-^  ^'  . 
and  therefore .  by  tqs.   i  and  2, 


I     n 

S  \  ' 


-f  <". 


» 


ind 


ClofK). 


•Ihc  (oUowms  Tabic,  calculated  by  Tutton.  gives  the  value 
of  the  backwater  huwtion,  </...,  .',  ,lu  ...s.  ot  a  dcuu  : 


1^1 ' 


290 


li.t(:KH  .-1  TiR   R  SCTION. 


■.tr 


^■•-A 


'^, 


o.ojo 

'f. 

.072 

.7.SI2 

285 

3860 

(.S 

"945 

.<H1I             3 

■1'37    1 

.074 

•7727 

.21^ 

381& 

f.9 

I<)|'< 

IK)2                I 

•9530    1 

.076       ! 

■7'>44 

.295 

3773 

70     , 

I '^92 

i)03           1 

.-iSl 

.<i7K 

.75''4 

.  31  li; 

373" 

71 

I5(,7 

■  W4          I 

.7425 

.o-o 

.748b 

•3"5 

3()-«.) 

72 

i?43 

CXJ5          I 

.64'*  5 

.0S2 

.7410  J 

.310 

3^.41, 

73 

ifep) 

.006          I 

.58S1 

■  "-4    ; 

•  733f' 

•3>5 

36.i.j 

74 

179^ 

.i*)7          I 

•5379 

.  1  >S6 

.72<J4 

.320 

357'    ' 

•.5     1 

•  772 

.ooS          1 

•4'>23 

.os8     1 

•7i'>» 

.325 

3532 

-() 

1749 

.  1  MX)                I 

.4539 

.ogo     1 

•7125  \' 

330 

.3495   ' 

/   / 

1727 

.010                I 

.4191 

.  <J<)3 

■  705S  i. 

•335     ' 

34?^ 

78 

1705 

.Oil                 1 

.3S77 

"94      , 

•('993  i 

•34" 

3422    1 

79 

11.S4 

.oii          1 

.35"'. 

09f> 

.f>i)39   i' 

•345 

33'S7    , 

So 

i()(,3 

.U13          I 

.3127 

LK)3        ! 

.*.866  ' 

35" 

3352 

81 

1(>42 

.014          I 

.3"^» 

IWJ 

.'''*"5   ■ 

.355 

.331' 

82 

I(.22 

.015           1 

2^55 

.l.)5 

'...55 

.3'« 

32S5 

83 

1(11)2 

.Old            1 

.2''44 

.110 

,1.1 

.365 

32^2 

84 

1583 

.017            1 

.2446 

.115 

'"7 

..37" 

3220    ' 

ss 

1564 

.i>iS          1 

aajS 

.  1 30 

.<)2  0 

•375 

3187 

86 

154'' 

...19       I 

.3081 

.125 

••'l.<9 

.3*0 

315' 

•»•" 

152^ 

.■>2()             I 

.1913 

.13" 

.(J034 

•3"<5 

3127    : 

Si 

■  Slo 

.nai           I 

■  1754 

•  «35 

•  59>3 

•3'J" 

3'«>7    1 

8.) 

14<l2 

.022         1 

1(>02 

.14"     i 

.5S07 

•395 

3068    1 

90     ; 

'475 

..23       I 

.1457 

•  145 

."S7"<' 

.4-KJ 

.•-"39    1 

9« 

145-t 

.-.14          1 

•  1319 

.15" 

.  5().  .S 

.41 

2.C3     1 

92 

M4I 

.<>S<             I 

.IlS(> 

155 

•5514 

■42        . 

292!* 

93     ' 

142? 

.i>2fl              1 

■  '"59 

.|6'> 

•5423 

•43        , 

2875     i 

94 

140,, 

'■•J7          1 

"93'' 

.  K.j 

.5335   . 

.44     ! 

2  8 -.'4 

95 

•393 

.<)3>*              1 

."817 

.170 

5251 

.45 

2774 

96 

>377 

.•.;.)          t 

.07-14 

■  '7  = 

•5lf'9 

■4''       , 

272f. 

97 

13'a 

.  sooo 

.47 

2680 

98 

1347 

•5ui4  . 

.4S 

2f.34  ; 

9<J 

'332 

-  J4     ,     1 

.Oll^l 

.  lijO 

•4939 

.49       , 

2590       I 

00 

1318 

.03.,          I 

.tx)07 

.«9$      1 

.4867 

.5" 

254*      1 

05 

lafo 

.03s 

•9S33 

.awj 

•479? 

.51       1 

25<lf)           1 

10 

i.«7 

.()4<i 

../.(..)  . 

.205 

■473" 

■52       ' 

24*'5          I 

15 

1128 

.042 

.9513 

2ro 

.4(,f,4 

■53       i 

342(1         1 

2' 

11.74 

...44 

.93''4 

.215 

.4(iIKJ 

■?4 

23>'*            1 

25 

1024 

.o4() 

1,223 

.  220 

•  45i>-< 

.^'l 

23?t           I 

'0 

"97  ( 

...4»     , 

,,,.■,7 

.235 

.447'^ 

•5<'       ' 

2314      > 

3? 

"93'i 

...?o      ' 

■ "')"-" 

.  2  -.CJ 

4419 

■57       1 

227')            1 

4" 

■■'u 

.''■  -' 

■15          : 

■  43<'3 

■5»       . 

2245             1 

1  = 

.Of  J 

1" 

.  |V<. 

■•9 

2213             1 

to 

.i>«fi 

.  •  > 

1  ,, 

J  1  -   1             1 

I  '7.^*  1 

•■'5*     ■ 

.■•4 

.'   1 

.■  1  r         \ 

•  175S 

.1X111 

. '" :"' 

4M5 

.I'2          i 

211(1            1 

1  J 

••72** 

.'.«ia 

.  !*.' 

.4o<)<i  ' 

'•3 

2'  'Sft            1 

7'  I 

I>?<)1) 

.iX)4 

.Si 

1  -I'^ 

'I 

-■ 

' '  "  1 

,<i<><i 

V,   ,  w 

1.; 

<MMl4 


B.ICKli',-1  Tl:K  /  t .VC//O.V. 


291 


i't) 


>■<)? 

.05 '4 

2.1") 

.ll5t)4 

2.  I 

.0527 

■1    1 

•  04')4 

2    3 

•"4"4 

2-4 

•<>437 

2-5 

.>>4I2 

2.6 

.038) 

2-7 

.03M 

2.S 

•  "34') 

2    ') 

.0.131 

•<.<i 

.0314 

.V4 
3-5 
36 


ciJ'iO 

-0 

.O07S 

20.1  J 

.fXlI  I 

n.'4-* 

^  .0 

.CXrfl2 

25<> 

OO.J7 

0237 

'J  " 

.l«)t() 

3"  " 

.01105 

021'* 

lO.'l 

....141 

35..) 

.0004 

■  )2IM 

1 1." 

.0035 

40.0 

.0003 

'ilhe 

12.-1 

.0'J3" 

4f  ■> 

.0«12 

"7- 

11." 

.0(>2() 

;;o.(> 

.(XX)2 

1(10 

14" 

.0022 

.19.0 

.oc«i 

1 4. 1 

15.0 

.<X«H)    ' 

liO.u 

.CKXil 

■  I  y> 

I'l.O 

.IOI7 

'J. 

.oxxi 

..I  i> 

17    '< 

>,"il'> 

III. 11 

!-•.') 

..»)it 

.1,.-., 

I'l.O 

.CK113 

N"IK.-    riif  vorresi>riiiil:..i;   T.ilili-.  liciliiccl  l-y  Urox-,  u  Iuim-  urk'iinicnt 
If  ■'r  y        .  '       ,„,iv  lif   :il    iincf  .■t.tuinivl    fr.mi    lliv   ;il"i\  e  Ijv  .i.hliin;   I 


,S  y^-^. 


II' 


to  riitli'ii'--   irminient. 

/;/  ///,  <,/.*•<■  of  a  fall.  Case  II.  .\rt.   1  ?.  in  tlic  limit 


aiul  llurifoic,  1)\-  0(1^.  .  1     and 


1      -r 
♦  3    '■ 


intt 


tan  4-' 


Tlu-  f..ll..«in<,'   lablc.  calculatiul  hy  "rutton.  kiv"  the  value 

if  tin   liaikwatrr  riiiutinn,  /      • 


39 


p» 


i 


|::.i 


2 

B.M.KUAM  R 

irscrios'. 

„ 

M~) 

\ 

<•,:! 

r       ' 

*(i) 

y 

*(-> 

ti 

11 

// 

' 

Ti       i 

„ 

...)■ 

1.5445 

.20 

•9?"1 

■45 

57?3 

.O'll 

2.7S7(. 

.(J42 

1.52:9 

.21 

•  93"3 

■  46 

?633 

.i)02 

2.5562 

.044 

1    5"7 

.22       ■ 

.9109 

•47 

SSI"; 

.<X53 

2.4207 

.  046 

1.4962 

•23       ' 

.S..22 

.48 

53M« 

.004 

2. 3244 

.043 

I-4S.3 

•^-t       i 

.8741 

.49 

5252 

.005 

2. 2497 

050 

I . 4670 

•25       1 

.851.6  , 

.50 

5167 

•  OlVl 

2.i8«5 

"5? 

••4335 

.26      I 

•''395   , 

<;i 

5054 

.01)7 

2.I3(>S 

.  t)tM  > 

1.4027 

■27 

.  8229 

•52     1 

494 « 

.008 

2.i>g2" 

065 

1.3743 

.28         ; 

.ho6S 

•53 

4829 

.od) 

2.0525 

.070 

I • 3479 

.29        1 

.79«u 

•54 

4717 

.010 

aoirr 

.075 

1.3231 

.30 

•7756 

•  55 

4607 

.on 

I •0554 

1             .080 

I • 2999 

•  3« 

,76ofi 

.56 

4497 

.014 

i.<>o3fi 

.085 

l-277> 

•32 

•  7458 

•  57 

43«i^ 

.016 

I.S<;.S4 

.090 

I    2571 

■  33 

■7313 

•  58 

4279 

.oiS 

1.81S5 

.095 

'.2372 

•34 

.7172 

59 

4171 

.1)20 

1.7827 

.  100 

I.21S5 

•  35 

•7'>33 

.60 

4004 

.022 

t.7502 

.  II 

1.1831 

■36 

.6S96 

•  65 

353'' 

.024 

1 . 7206 

i          -'2 

I • 1 504 

•37 

.6762 

.70 

3019 

.02'" 

1.6036 

•  '3 

I. 1201 

.3S 

.6629 

•  75 

2510 

.<»2S 

1.667S 

14 

I. 0918 

39 

.6499 

.80 

2(5o4 

.030 

I  6441 

■15 

I. 0651 

•  40 

.6371 

.90 

1 001 

.  (132 

1 .6219 

.16 

1.0399 

•  41 

.6214 

1   -O*.! 

(RXK) 

•"3« 

I. 6010 

.17 

I .0160 

.42 

.f.119 

.  1  >3fi 

i.5l<n 

.!■< 

■193' 

•  43 

•  5995 

.038 

I  .  562(> 

•  ••) 

•0713 

•44 

•  5S73 

i 

\,,ii  . — Krcssc  uses  tlu'   sanR'  value  —  .i^cKxj  tor  the  oni- 
staiit   I    Ix'tli  for  a   dam  and  for  a   /  •  7       WU  arj^uinent  in   tlie 


latter  ca.se  is  '  1  ,',..•.:■.■!   t..   ,,l)tain  Hresse's  T.ihle 


//■ 


fnim  tile  above,  the  ar^nmient  adopted  by  'luttoii  is  suhtr.utrd 
from  I,  and  .i>04C)  from  the  value  of  (/>i;i. 

l)ui)uit,  ai^.iin,  uses  tlie   ar},'unient   '       ami   lijs    Tables   in.i_\- 

bo  obtained  from  those  ^iven  b\-    I  utton  by  ciiiiatiiii;  his  Daek- 
water  function  to 


md  t. 


1 .41  5.S    t-  ),  —  '''  •"'  '"'■  -i  '■''•■• 


2.0204  -    '..  -  <M.:)  for  .1  fall. 


CHANijF.    OH  CROSS-SECTION. 


293 


Dupuit    ncL^'U-a-^    the    term 


.uiit    iiuhidr^    ill    lii--   liatk- 


watcr    luiiai..n.     uhuh    may    l.c    .Icsi-natcd     '      ,     tlic    t>-nn 
,.  _  „'  [_-L~-'' j  i„  c<iiiali<in     ^.  ■-"  that  hi-  t. .rmula  h.'cnnu-; 


// 


=  /!:;i  -  /(O.oi ', 


,::  —    .01,  ai  (.ufati:    mca-^iircnicnt  i-^ 


i.(.nsulriin;4  that  when  '^^ 

nil  hiniicr  jio'^sil.lr.      Kiihhiianii  ^ivcs  tin/  same  rule. 

17?  Change  of  Section.— Cask  I.,  I'i,^'.  i6n.  A  cliannd  of 
slope.',  an.l  in  ^^huh  the  How  is  steady.  <,n-ailiKilly  c.Mitracts 
IV.MU   a  uMth    .1-1    -    /■■,   to   a   ui.lth    rc'=  /-', ,    the   surface   of 


stca.lv  motion  l.ein-  IV  above  .1.1.  and  A' s  l-elow  CC.      On 

appr.iachin-    ././    the  surface   t,'ra.lually   rises  ami    reaches  its 

-roatcst   hcifrht    Q T  =  ::  above  /'(,'  at  ./.I.      Tliis  is  followed 

b\  ,,  -ra.lual  fall  to  the  surface  of  stca.ly  motion  A'.S'  at  (  t  . 

1  .,   //  ,  //    f  -.  Ii  \  he  the  depths  c<irrespondin^;  to  steady 

lu.itiMii    above    AA    and    below    />'/>, 

rcsjicctivoly. 

"   f//,  .  III.  be  the   mean   hydraulic   depths  ab-.ve  ././  and 

below  /.'A',  respective!)  . 
•'  «  .  //,  be  the  mean  velocities  of  flow  above  A.\   and 
below  />7A  respectively. 


294 


ai.-1\(J/-:    Ol     '-A'C).s5-.S7;V;r/().V. 


'ri.iii,  (lisrc^%'inlinL;  tlic  ctTcct  <>(  surface  rcsist.iiKc  hctWLH  ii 
.1.1  .iiul  ( '( ', 

'  I        '        ->  .  f-  -  t  ,r 


or 


.:   rzr   //  , 


"."   —    ",' 


li  the  scctii'ii  i-;  ,1  ICC.  tanL'k', 


Vx 


pl-4 


//|/>|//|         lijl.n.. 


lUit 


'IhiTL-forc 


aiul 


\    \       /'_,'"/_,  /  2,1,'- 


It  tilt.'  width  is  j^rcat  as  toniparc.l  with  the  dcptli, 


w, 


'I  lurcfore 


=     '     aiuI      w/j  =    ■-,  appr. >\im.itcl>'. 
tt.,'        Ill,  ^    It.,  _  /-',//,         ■)  //>',•' 


LHASliE   Oh   CROSS-ShlCTIOS. 


29  = 


C  \.|  1 1 .  A  .1,,-inncl  of  slope  -.  in  winch  the  flow  is  steady. 
/\)A'.S  iHU.-  the  surface  of  steady  nrntion,  ^^radually  contract^ 
from    a    width    J  J  =  /',    to    a   narrouer  width    at   CC       Mu 


Fic.    i''i- 


Channel  rcnains  narrow  for  a  limited  distance  (7^  an,l  then 
oraduallv  enlarges  to  its  on^mal  s,.e  at  /:.  <  )n  an-n.u  hm^ 
,    ;  tl,..'surf^ice  rises,  attain,  .ts  ^reatc.t  hei-ht  (jl  ab..ve  /  (j 


A. I  the  surface  rises. 

at  J,  falls  to   I'  at  (  .  then   i^  ,1   point 

tH.m    \['  tM   the    -iirface    o!    steady 


IC  below  /V'A'N  at  /', 


and    fuialh-    suddenlv'    ri- 
ni  ition  at  A'. 

Let  ~  be  tlu    depre-^-^ion  ot    H' below    /'.S. 

..   /;,    /,\  be  the  uidth.s  ;.t   /'  anil  A. 

••    i(,  11^  be  tne  mean  velocita-  at  /'  and  A. 

Th.n 

li^  —  «,■ 
>" — -'      • 

where  <if  may  be  taken  =   l.i. 
If  the  section  is  a  rcctanj^le, 

Therefore, 

a  cubic  e(Hiation  givin.L;  .:. 


2  9' 


CH.-ISiU:    Ol-   CKOSS-SFCnOW 


p::i 


1  he  ■^urf;n:i'  /'/:  ni.i)'  now  be  plotteil.  and  ('7'  nKi\'  be 
iounil  as  in  Case  1. 

These  expressions  also  ^ive,  appmximatel}-.  the  ciepressjon 
l>e'n\\  tile  surface  /VVv'.S  of  steatl\'  motion  when  tlie  ihannel 
has  il^  seitioii  suddenl}-  ehanijetl  by  such  ubstructiun.s  as 
briilije-piers. 


On  approaching'  the  pier  ends  the  water-surface  f:;ra(luall>' 
rises  to  the  nia.xiinuni  hei_L,dit  /'  abo\e  /'S,  then  falls  to  AT 
below  /'.S  between  the  ])iers,  and  fni.illv  rises  attain  to  the  sur- 
face ot  steady  motion  on  ]),issinL;-  into  the  open  iliannel. 

Let  /-'i  ,  /)'  be  the  distances  between  the  axes  ,nul  the 
inner  f  ices  of  the  piers. 

Let    //  be  the  .lei)th  iiehiw    .\T. 

Let   J  i)e  the  fill  from    /  t.)  .V. 

'I'hen,  accordini^  to  Hresse.  the  \altie  of  a  is  Ljiven  b\-  the 
empirical  formula 

(,    beinij  a   coefficient  of  contraction   and   ha\ini;    an    aver.i^^e 

value   of  about  .><.      Also.   (J  is  the  discharge  for  the  width  />', 

ol'  the  channel. 

*   This  (nrimila,  altlmuKh  );ciKT;illy  adoplfd.  is  optrn  to  qufStion.    IJrtssf 
coiisidfrs  tli.it  ail  i'<|iiallv  Cui  reel   a]i|iti)xiination   is  olitaiiied  at  a  distance 

i)f  21)  /•',    -    /.')  from  llic  tuntractioii  by  taking  ;     -  mi/it     '  _  j)  . 


c,.ii:ci\G. 


297 


18.  Gauging  of  Streams  and  Watercourses.      Ihc  amount 

(,f  lliiu  ('  m  cuijK-  ICLt  per  sccoiul  .icrds-,  ,1  tr.in  .viT-.c  scctinii 
of  J  s(j.  ft.  m  area  is  <,nvcn  by  the  expression 

It  \w\n\:,  the  nic,in  vcloiit\-  ..f  flow  in  tin-  -action  in  feet  per 
■-econil. 

If  the  lonyitiKhna!  profile  am!  several  transverse  sections  ot 
a  ihannel  can  be  plotteii.  the  volume  of  tlou  nia\-  be  calculated 
by  me.uis  of  eip   '  1  1,  p.   -7?- 

\x\.!i    .11 /,^  be  the  mean  xelticitie--,  . /,  ,  .!_,,   .   .   .-'„ 

the  areas,  ami  /',  .  /',,..■  /'„  the  wetted  pernneters  of;/  sec- 
tions of  the  channel  at  the  si)ecified  distances  /,,/,,.../„_, 
apart.  Then  .~,  the  fall  iu  tlie  free-surface  level,  which  may 
be  found  from  the  lon^^itudinal  profile,  is  -^iven  by  tlie  equation 


21"-  I      III  -A' 

.^  to 


in  which 


and  «  m.iv  be  t.iken  :=   I  .  I , 


lUit  .l,.v,       A.;  --  (} 


A 


'-■',.    '""l     '"  =y.- 


'lluTc'fore 


-^.A  I  i       A  •-•/        2i'-  /     A' 


and  0  can  be  calculated  as  soon   as  the  integration   has  been 

effected,  uhicli  may  be  possible  if  /'  and  .J  are  known  functions 

of  s.      .\n    ai)pro\imate   vahu;   of  the   intei,^ra!    may  be  found 

L;r;iphicall\'  as  follows; 

/» 
I'lot,  as  ordinates,  the   values  of      ,  at   the  //  sections,  and 

join  the  upper  ends  of  those  ordinates.       The  .irea  between  the 


29.S 


li.-uaiM;. 


t:--^ 


fXtrcMiu-   ordiiiatcs,  tlu-  axis,  ami   tlic   line   tliiis  (ietcrniiiied   is 
tin.-  \ahic  requirfd. 


Fig.  163. 

riencrally  s]K-akinj,'.  liowcvc-r,  the  above  mctliod  of  gautjinf; 
till-  flow  of  a  stream  is  not  vcr,\-  accurate,  on  account  of  the 
errors  in  tlie  vahies  of/',  ./,  and  tlie  intcj^'ral.  More  correct 
results  are  obtained  b>-  determining,^  the  mean  velocity. 

19.  Determination  of  the  Mean  Velocity  u.     Mi  iihm'  I. 
'IIk'    nio-^t    convenient    method    lor    .:.,.iUL;inj4    >nial! 
streams,  canals,  etc..  is  I)\-  means  of  a  temporarily 
lonstnictcil  weir,  which  usually  takes  the  form  of  a 
rectanj,'ular    notch.      The   j,'reatest    care   should    he 
exercised  to  insure  that  the  crest  of  the  weir  is  trul\- 
level   and    i)roperl\'  formed,  and   that   the   sides  are 
trul>-  verticil.      The  difference  of  level  between  the 
crest  <if  the  weir  and  the  surface  of  the  watir  at 
])oiiu  \>  Ik      it  has  not  bet^un  to  sjope  down 
the   V  eir   is   best   estimated   by   mians   of  Hoyuen's 
hook-i^au_L;e,   Iml: 


a 
ards 


104. 
This    5,^'lU5^e    consists    of    a    carefull\-    ijraduated 
roil,  or  of  a  rod  with  a  scale  .ittached,  havin^r  .it  the 
lower  end   .1   liodk  with   ,1  thm  Hat  bod_\-  .iiul  a  fine 
l)oint.        The  rod  slides  in  \ertical  supjiorts, 
anil  a  >low  \ertical  movement   is   t;i\en   by 
me.ins   of  a    screw    of  fine    pitch.      .\    stiff 
vertical    rod,    with    .1    sh.irp   point,    havir.g 
been  i^laced  at  5   to  S   ft.  from  the   back  of 
the  weir,  w  ith  the  point  on  a  level  with  the 
weir  crest,  w.iter  is  run  into  the  flume  until 
it  rises  slij^htiy  above  the  crest,  producing 
a    capill.ir)-  elevation   .it    the    point.       i'he  Fig.  164. 

water  is  then   .lUowcd  to   subside  until  this   elevation  is  barely 


(i.-iLaisa. 


209 


licnoinible,  when  tlic  rod  may  be  removed.  A  hook-K'ai'K'e  1- 
next  placed  in  tlie  same  position,  and  the  hook  is  slowly  raised 
until  a  capillary  elevation  is  produced  over  its  point.  1  he 
li.),,k  is  then  sl.iu'y  lowered  until  the  clevadon  become-,  almost 
imperceptible,  when  a  readin-  is  taken  correspundin^'  t<i  the 
level  of  the  crest  of  ti.e  weir.  M'>re  water  now  flows  into  the 
tlume  and  over  the  weir.  .\s  s..,,n  ,is  the  motion  h.is  become 
steady,  the  hoi.k  is  raised  an<l  the  jioint  adjusted  at  the  surface 
in  the  manner  just  described.  .\  s-cond  readin-  is  taken  an<! 
the   difference  between   the  two  readinj^s  is   the  head  of  water 

over  the  crest. 

In  ordinary  li-;lu,  .liffercnces  of  level  as  small  as  the  one- 
thousandth  of  a  foot  can  be  e.asily  detected  by  tb.e  hook- 
<;au-e,  while  with  a  favorable  lii,ht  it  is  said  that  an  experi- 
enced ..bserver  can  detect  a  difference  of  two  ten-thous.m.iths 
of  ,1  foot.  Such  differences,  however,  cannot  be  measured 
under  the  ordinary  conditions  of  practical  work. 

Ml  111.)!.  II.  A  p.)rtion  of  the  stream  which  is  tolerably 
straight  and  ..f  ai)i)roximateK-  uniform  secti.ni  i-.  defined  by 
tu./tran^xerse  Imes  (\A/;,   (\C'/>  at  any  distance  .S  ft.  apart 


Fi<;.  I*);- 


rhe    base-line    <>,(',  is    parallel  to    the    thread    /■/'  of  the 
nil   observ     ■•  with   chronometers  and   theodolites    or 


stream,  a 


300 


(i.-ll  lilSii. 


sc\t.iiit>  MX-  statiiiiicil  at  (',  ,  ( ',.  I'Ik'  time  /' and  p.illi  /:'/•" 
taken  li\  .1  li'ial  lictuccii  .  I  A'  ami  ( V-'  can  now  he  ilctci  mined. 
At  the  numicnt  tlie  float  leaves  .1/1  the  observer  at  ( ',  M;^nals 
the  observer  .it  ( '., .  wlio  measures  tlie  an^de  ''.'V-.  -'"d  each 
marks  the  time.  On  reacliini,'  <  V^  llie  observer  at  f\  sJHuals 
tlie  ob-^erver  at  (\.  who  measures  the  ant^ie  (l.<\/'.  and  eaiii 
a^'ain  marks  the  time. 

|-".xj)erience  alone  can  i^uide  the  observer  in  nxup^  tin  dis- 
t;nue  .S  between  tln'  points  of  oi)servation.  It  sh<iuld  be 
remembered  that  altliouL^h  the  error-,  of  time  ob>er\  atioiis  are 
diminished  b\-  irKreasin-,'  .S,  the  errors  due  to  .i  deviation  Irom 
line-,  parallel  to  the  thread  of  the  stnam  arc  iiurease<l. 

A  number  of  floats  ma>-  be   sriit  aloiv^  the  s.ime  [Mth,  and 

tin  ir  velocities  f  '..  j  are   otten  found  to  vary  as  muih  as  jc.  per 

lent  and  even  more. 

I  lavin:^  thus  found  the  \ilocities  alon:^  an\  required  number 
.f  p.ith'  in  the  uidth  of  tlie  stnam,  the  m«  an  \eltnit>  tor  the 
uliole  widtii  can  l)e  .it  once  tletermined. 

SurtiiK  -lli'iits  are  sin. ill  piices  of  w.iod,  cork,  or  balls  ot 
u.iN.  hollow  metal  and  w.iod,  eolored  so  as  to  be  clear' y  seen, 
anil  ballasteil  sous  to  flo.it  iicarlv  flu-h  with  the  \\.iter--urtace 
iiiid  to  be  little  alVected  b\-  the  wind. 

Sii/'siir/,ii,-ffi>iUs.  .\  -ubsnrf.ice  float  consi-.t^  o|  ,i  hi  .ivy 
flo.it  with  .1  conip.ir.itivel)-  l.u:4e   intercepting'  are.i,  maintained 


fi1  r^. 

Tin    \<'(<  1  •   •    •     • 

at    .uiy  itiiinied    depth    bv  mean-    ■•''    •   -■•>    '"' 


OAi'iilS'i. 


,"' ' 


wi-tical  innl  attached  t-.  a  -uilal.ic  surUuc-lloat  ..t  .uu.ium.u. 
uiumTMcm  aiul  resistance  i-'i-.  K-o  sl„,vvssi.ch  a  conil.ina- 
ti,m.  tlie  !M«ir  lluat  cmsistir^'  .-Itu..  i.ie.es  <.f  .ualvani/e.l  imn 
.,,l<!eied  t.-.-e;'iu.T  at  ri-lit  angles,  tin   ui-pei   tl.ul  l.cin-  iiicrely 

a  wooikn  ball. 

Annil'er   LonibiiKiti.-n   .-l   a   h-ll-.u   nu"  •.!  '..all  uilh   a   piece 

..r  cork  i-  -liMUM  by  lii;-    "',"• 

\\w  in.iiinii  ..llhe  iMmbination  is  siiisilily  the  saiiu-  as  that 
..\  ilu-  s„b!nerL;c-.l  ll-'at.  aiul  -ive.  th.-  velocity  at  the  .leptii  to 
AliicJ!  the  hea\y  float  is  siii)i:i-rKe  i- 

l:.-iii-lf.'„l>.—  \\\'^  e-iual  aiul  similar  floats  I-i-.  loS^,  one 
.  ii~ir  ami  thi'  oilur  '  ■  •"'"^ —    — 


.nnecteil   he  a  tine  coil.       1  ^'x  \>   o  i;\   ■  :•,) 


of 


/ 


;iic  combination  is  iijiproximately  tlu'  mean  of 
the  surface  velocity  '  r   i  and  of  tin    \clocity  •.-,,• 
,1  t!u   del)!!!  to  \^llich  the  heavier  float  i-  sub- 
nerved.       Tlui- 


An^\  tlierelore 


Ki.;.  t'.S. 


-o   th 
biill 

; 


.,t    .     ..m  be  .lvt«rmined   as  -oon  as   the  value   of  ; ,  li.is 
..bscrve.l  and  tlie  value  <.f  .-,  found  !)>•  surface-fl(»ats. 
',!.'.  itv-iod.  -    rilis  i-  a  liollow  cylindrical  rofl  of  adiustal)ie 
len-th  and  ballaste.!  •  float  marly 

\crtical.  It  sinks  aiiii.-l  to  the  be«i  ot 
the  stream,  and  its  velocity  .• ,.,  i  is  ap- 
prosintately   th<     m.an    velocity    for    the 

whole  <lepth. 

Ir.incis  141VIS  the  lollowm^  tnipirual 
formula  connectin;;  the  uir.m  \ .  locily  > .  „  1 
.eith  the  <  bxerved  veloii'  '  the  rod: 


In;    l'i<). 


I  < '  I  J 


\ 


'■'A 


,/  Ihuil;  (k-i)tli  of  striain,  ami  </  tlie  (lc])th  of  water  below 
imttoiii  ol  pi(l ;  hut  (/  -ihoiiiil  not  cxccctl  al)i  iiit  one  t'lmtli  o|  </. 
Mr.l  lltiD  111.  /',/,'/  / ii/>f  ,vi,/  />-tny  ddiiX'-  —  -'^  I'ltottiilje 
I  I''ii;s.  i-()to  \-2  m  it--  simple'st  furin  i-^  a  _i^la--s  tul)e  u  itli  a 
1  i.;]it-an;4k(l  Ix ml  Wlu  n  the  tiihi-  is  ])kini,'eil  \ertically  into  tlu' 
•-'.naiu  to  an\  ieiiiiiii-il  ileptli  lielo\\  the  free  Mirtai<-.  with  it-. 
in.. nth  iMiintinL;  \ii>"triani  ami  normal  tn  the  direction   df  IIdw. 


Fp;.  171'.  Tii;.  171  Fn;.  17? 

liii  A  .iter  ri>r-  iM  lie  uiMi  10  .1  liei^^lii  /,  loi  v  I  iiic  <iiit-ule  ■surface, 
aiiil  the  wei-ht  oi'thc  lolninn  of  water,  *  ■  li  hi_L;li,  is  halance"! 
i)\- the  impait  ■■I'tlie  stream  nn  th<'  mouth       llenco    Chap.  V) 


and  tinrefort- 


^4' 


//  =  /( 


(►-;  . J 


./  ln'in^  the  sectional  area  of  tiie  tube.  11  the  velocity  of  (low 
at  the  j^iven  depth  ■•'■!  '-  a  eoeffKieii;  •  '  determined  t)V 
«  \i<erinient. 

\  mean  value  of  1  1  1,  With  a  funnel-mouth  ^r  a 
liell-niouth  I'itot  found  ■;  to  he  1.3.  I'his  form  of  mouili, 
howe\er,  interten^  with  the  stream-lines,  an  1  the  \tloiity  in 
front  of  the  m<iuth  is  prohahly  a  little  different  from  that  in  the 
uiuihstrueted  -trc.nn. 

!  he  .iihantai;e<  of  tubes  of  small   -ection   are   that   the  dis- 


CAL'CISG. 


.;o.> 


of  tlic  coluir  .'  w  ator  arc 
ilitAkcd.  I)ar».\-  foiiiKl  Ijy 
I  aictul  mca-iuicmciit  that 
tin  (litTtTfiKc  of  level  bc- 
lu(-cn  the  suilai<"-  "t  the 
>\attr-C((lumn  in  a  tube  nt 
-mall  section  plaeetl  a>  ni 
l-i;^.  170.  and  of  the  uater- 
Innni  placet!  as  in  V'v^. 
:  71  w  it'-  its  mouth  parallel 
to  the  direction  of  (low  ,  is 
almost    cxactl)-    equ.d    to 


W  lull  the  tube  is 
placed  as  in  Fi^.  1  72  u  ith 
it-  mouth  pinntinii  down- 
stream ami  normal  to  the 
direction  of  flow  .  tlu-  level 
of  till'  surface  of  the  water 
in  the  tube  is  at  a  deptli 
//■  below  the  outsid  "•- 
face,  and 


where  k  is  n  ci'eftlcient 
to  bi  determined  by  ex- 
periment and  a  little  less 
than  unity. 

In  this  caHc  the  tiilx- 
aj^jain  obstructs  thistreain- 
lines.  I'it'it's  tube  does 
not  ^ive  measurable  iiuli- 
cations  of  vtT>"  low  veloci- 
ties.     A  serious  objection 


£ 


1:   .  .1 


J 


I       1- 


f::4 


304 


GAUGING. 


t.  the  simple  I'itot  tube  is  the  difficulty  of  ohtainiiiii  accurate 
readiiit;s  near  the  surface  of  the  stnain.  I'his  nhjection  is 
nnioved  in  tlie  case  <.f  Harcy's  t^auge,  sIimwii  in  tl.e  accom- 
jianxin;^  sketch,   I"i^-  173- 

./  and  /.'  are  the  water-inlets;  ('  ami  /'  are  tun  double 
tubes;  A  is  a  brass  tube  cont-nnui;  two  -^dass  i>ip<'s  wiiich 
(..nununicate  at  the  bottom  with  tlie  water-inlets  and  at  the 
top  uith  each  .)ther,  and  witii  a  pump  /'by  which  tlie  air  can 
be  drawn  out  of  the  ^l^ss  i)ipes.  thus  aliowini;  tlic  water  to  rise 
in  them  to  any  convenient  heii^ht. 

Thus  Darcy's  i,Mu^'e  really  consists  of  two  I'ltot  tubes  con- 
nected by  a  bent  tube  at  the  top  ami  havin-  their  mouths  at 
ri^ht  ant,des  or  pointing'  in  npi)osite  directions.  If  //  is  the 
di?ference  of  level  between  the  water-surfaces  in  the  tubes 
wh   n  the  moutlis  are  at  riL;ht  an-les,  then 

->  ,r 

and    Darcy's    experinu  nts   indicate   that   ^'  does   n.>t     s.r.sibly 

difier  from  unity. 

When  the  mouths  pomt  in  opposite  directions,  let  //, .  //_. 
be  tlie  diflerences  ol  level  between  the  stream-surface  and  the 
surfaces  of  the  water  in  the  tube  i)ointin-  uj.-stream  and  the 
tube  puintinti  down-stream,  respectively.       1  hen 


/(■, 


-.< 


aiiil  till  rcforc 


11 

mi' 


^■|+'^":^ 


2ff 


w  here  I:  —  l>\  -\-  k^. 


^  h 


G.-IVGIS'G. 


305 


/!  liaving  been  (lotermincil  c\ponincnt;ill\-  once  for  all,  the 
liiHerence  of  level  (=  //,  -;  /r,\  between  the  columns  tor  .my 
;^'iven  case  can  be  n\easure<.i  on  tiie  t^Mui^^e  ami  the  value  of  11 
^  .m  tJKn  be  ImuiiiI. 

A  cock  may  be  inserted  in  the  benil  conneaui-  the  two 
tubes,  .md  throu-;h  this  cock  air  m.iy  be  exhausted  and  a  par- 
tial vacuum  created  in  the  upper  portion  of  the  ^'au^^e.  1  h.- 
water-columns  uill  thus  ri>e  to  hi^dier  levels,  but  the  dilference 
between  them  will  remain  constant.  Thus  the  surface  of  tiie 
eoKunn  in  tiie  dow  M-stream  tube  ma\-  be  brou^r|,t  .ibove  the 
level    of  the   outside   surlace.    and    the   readin^^    i>   tiieii    easily 

IlKldr. 

Sometimes  the  i^au^'e  1-  tuniisheil  witii  cocks  .it  the  lower 
parts  of  the  tul)e^,  and  il  these  cocks  .ire  closed  when  the 
nie.isiirement  is  to  be  made,  the  -au^e  m.iy  be  removed  trom 
the  -tream  for  the  icadinijs  {,,  he  taken. 

Ml,illii|i   1\'.    (  iiiii  iit-iiiitin.  —  The    velocity   of   flow    in 
lar.Lie    streams    .111  i    rivers    is     most    conveniently    and     mo-^t 
.iccuratclv   a-.cert.niud   b\    means  of  the   current-meter.       I  li  • 
<  .irhest  form  of  meter,  the  Woltmann   mill,  is  merel\-  a  water- 
mill  with  flat  v.me-..  similar  in   theor>-  and   action   to  the  wind- 
mill      When    tlie    Woltm.mn    is    plunj^ed    into    a    current,    a 
counter   re'^'ister^  tiie   number   of  revolutions   m.ule  in  a   -iveii 
interval   of  lime,  an>l   the  corresiH)iulin,L,'   velocity  cm   then  be 
«letermined        I'iii-.  f>)rm  of  meter  lias  ^'one  out  of  use  .md  lias 
been   replaced   b\    ,1   v.iriety  <'f  meters  of  ^Tcater  accuracs ,  of 
finer  construction,  and  much  better  suited  to  the  work        in  it-, 
simplest  form  the  present  meter  consists  of  a  screw-piop.lK  r 
wheel  '  n^'.   174).  or  a  wheel  with  three  or  more  vane^  mounted 
on  a  spindle  and  connected  by  a  screw-jrcarinu  ^^th  .1  counter 
which  reL;isters  the  number  of  revolutions.       The   meter  is  put 
in  or  out  of  K'ear  by  means  of  a  string;  or  wire       When  .»  cur- 
rent velocity  at  an\'  L;iven  jioint  is  to  be  found,  the  re.idm^;  of 
the  counter  is  noted,  the  meter  is  sunk  to  the  reijuire.l  jiositini. 
.iiul  i-^  then  -et  and  kept  in  gear  lor    my  specified  interv.iJ  of 


3o6 


a.a  (,isi,. 


tiiui  At  tin  mil  .it  tlK  iiitcrxal  the  meter  i'^  put  nut  of  ^'car 
aiul  I-.  raised  to  the  suiiate,  when  llie  readiiii,'  of  the  c<>ii;iter  is 
a-aiii  noted.  'Vhv  <UtTerence  hetweeii  tile  readuv.;^  L;ives  the 
number  of  revoUitinns  made  durin-  the  interval,  and  the 
velocity  i^  L^i\eii  by  an  empirieal  t'ornuila  eoniicctin^  the 
velocitv  and  the  numl)er  of  revolutions  in  a  unit  ot  t.iiie. 

I'he  vane  /'is  introdueed  to  compel  the  meter  to  take  a 
direction  peri)endicul.ir  to  that  of  ihr  stream-lines,  but  thi- 
iiia\'  not  necessaril)' be  perpendicular  to  tlie  axis  ot  tile  stream. 
'I'iic  sliL;lit  error  due  to  this  discreiiaiux'  is  usuall>  disreL^ariled 
ill  jiracticc. 

In  oriler  to  prevent  the  inecli.inism  of  tlie  meter  troin  beint; 

I  hi.    174. 


■lp= 


p:-4 


Flo.   175. 

iiiinrioii-i%  ,,iiecte<l  by  lloatitii,  particles  of  detritus,  Kevv 
cncldsod  the  counter  in  a  brass  bo\.  l-'i-^.  i;?,  with  a  .L;lass 
t.ice.  .iiid  filled  till'  i)o\  with  pun  w.iter  so  as  to  in-aire  a  cnii- 
staiit  coefticieiit  of  friction  fi>r  the  parts  vvliich  rub  .i^'aiiist  each 
iitlicr.  Ill  the  best  meters,  iiouever,  the  record  of  tlir  number 
«»f  rtvtilutifHis  is  I.cpt  tiy  means  of  an  eli-ctric  ciruiit,   lii;.   l^f). 


c.-ii  (,i\<;.     — 


.lO; 


'.vhioli  i-^  111, uU'  .111(1  hriiki'ii  mikc,  or  niorr  frc(]iic;itly.  cull 
rcvolutiuii,  ciiul  whii-h  .ictu.itc-^  tiic  icon  iliiii;  a])i)ar.itu-.  I  he 
time  at  wliicli  an  cx])cniiuiit  l)i'L;i!i->  and  cmls  is  noted.  <uul  the 
revolutions  made  in  tl;  ■  interv.il  are  read  on  the  counter,  which 
m.i"-  be  k--])t  in  a  hoi.L  or  on  the  shore,  as  the  circumstanci's 
>)(  the   case   ma_\-  re([uire.      'ilu    meter   is  usual!)'  attached  to  a 

Ku..  170. 


Tl. 


'iuitahlv  ^'raduated  ]»i\r.  so  tli.it  the  d.pth  of  the  metrr  helou 
iht  water-surface  can  he  directl\-  re.id.  In  deep  .md  r.i|)id 
uater  tlie  meter  must  he  he!>l  by  a  wire  cord  whicli  will 
.sualh-  require  to  hv  ^u\ed  to  .1  forward  line.  I'lie  mean 
velocitv  for  tile  wliole  deptli  at  aiic  point  of  a  streair.  m.i\-  l>r 
found  by  mo\ini,'  tlie  meter  vertically  down  and  tlieii  up,  at  .1 
uniform  rate.  The  iiuaii  of  the  readinj^s  at  the  two  surface 
iiositinns  and  at  tlic  bottom  position  will  be  the  number  ol 
revolutions  corresponding'  to  the  moan  \elocit>  reciuireil.  Ilu 
mean  vclocit)'  for  the  v\  liole  crriss-scction  ma\' also  be  deter 
mined   bv   moviii'^   the   meter   uniformly   over   ail    parts   of  iju' 

section. 

I  he  meter  siloiild  be  lated  Itolli  i)efore  and  .ifter  it  is  iis(  d. 
This  is  clone   1)\  drivini.,'  tlie  meter  .it   different  uniform  sp,.,ds 


3oS 


(i.U  tjisr, 


!::•» 


tlirou-h  ->til!  water.  I,-  Doriinciit  >h<.\\.  that  the  vcir.uty  r» 
;..!h!  tlie  number  of  levoliuions  n  are  approximately  connected 
by  the  t.  irnnila 

7'  =  an  -f-  /', 

where  .r  and  /'  are  coefficients  to  be  (leterinnied  by  the  nutho.l 
of  lea-t  s(iu.ires  or  otherwise. 
i;xner  :.;ives  tlie  formula 

r-  =  <■■«•'  +  T-,,-. 

7-,  b(  in-  the  \eliKity  at  uiiich  the  meter  just  cca^^es  to  revolve. 

(diiiK  Ml  1  iiMiiS. --M  in>  "ther  pieces  of  apparatus  tor 
ttie  measurement  of  current  velocities  have  been  ilesi-ned. 

I'errodil's  hydnHlynamometer,  for  example,  -ives  the 
velocity  directly  in  terms  of  the  an-le  throu-h  which  a  vertical 
torsi.in-rod    i->   twiste.l,    and    in    tliis   re-^pect   is   superior   to   the 

current-meter. 

hydronutric  pi-ndiilnm  I'i^'.  i"''^*, 
aj^Min,  conneit--  the  \elncit\'  with 
the  ani^ular  deviation  from  tlic 
vertical  ot  a  heavy  ball  suspended 
by  .1  -trini;  in  the  currint. 
^  I  hihometric       and       torsion 

bal.mces  have  also  l)cen  devised, 
but  the\-  nnist  be  re<,'arde(l  rather 
as    cnrio-~ities    than    as    bcinj;    of 

any  real  i)racti'  al  u-e 

ll.ivin'4  found  tile  maximum  surface  velocitw  :•, ,  at  .my 
i)oint  in  I  watercourse.  b\  one  of  the  above  methods,  then 
<Art.  lo.  p.  .?;<)-  tiie  mean  \elocit\- of  the  whole  section  is 
},'iven  b\-  the  empirical  relation 

If  the  transverse  section  of  the  water\\a\,  at  the  point  in  nees- 


The    tachometer    or 


Kic.  i-s. 


GAUi.lNG. 


1^9 


tion.  is   plotted   ami   its  arc.i,  j 


mav 


be  at  once  ca 


1,  measured,   the   di<Jiar-e,   (j. 

1,1 


kulated  hv  means  of  the  tnrnniLi 


Q-\Ai 


Auain.  sdretint^   an  approxnnatel) 


1\-  'tr.ii'jlit  length  (.t\li.innel 


let  .1   be  the  distance  Iron 


the  ori.Ljin  of  a  particle  in  tile  surh 


lil.unen 


particle  is 


t    of    inaxinuiin 


.il\ 


.It' 


;ini 


therelore 


>  =  An  -  -.\ 


rhni    the    veloci 


Ix 


t\    of  this 


4     ''■«• 


,ir 


Hence 


C"  =  ^:/ 


A.i.x 


t  beinj^'  the  time 


in  uhich  the  tloat  passi  • 


.er  the  ili>tance  i. 


If  this  distance  is  now  ili\ 


(led  into  ;/  eiiiial  divisions,  aiu 


J„.  J,.   ./,.  -■' '-. 

commencement,  at  the  \n 


tl 


ic  ,in  ,1--  < 


,1   the  u,iter\\a\-  a 


I  it 


t  tlu 


1  I  intermctliate  division  points  a 


nd 


at  the  eiul 


the  UiiL/th  A.  then.  b>-  Simp- 


i.n  s  rule 


,/ 1 


\H   I 


.K+2i.l,   ■    .(, 


-|-4,J,4--J.>+-  •  •  +  -'' 


IV 


The  inteijration  m.t\ 
f  . 


d'-'i  be  at  once  e 


ffected  if  .1   is  ^MVen  as 


junction  ot  .» 


Aj,'ain.  if  //  i-  the  d.  pth  "f  -t. 


,id\    iiiotion. 


hii 


-,»  _ 


am 


1    il   the  V.  ii 


1th 


,f  the  ih.mn.l   is  lar^c  as  compared  vvr 


th 


//,  m        //,  api>roNim.itely.  aiK 


1  . 1        />'//. 


3'f 


GAUCIS'G. 


1  hcioforc 


Q  --=  I 


\/:"' 


At  any  ^nw  n   jx.iiit   iti  tin;  stn-ant    /•'  in  i\-  he  considered  con- 
'•tant,  /in  aNii  ciin'>tant ,  am!  a  inotTuit'iit  m  !Tia\' hi'  sul)stituti_'ii 


)<.r  /.' 


\/; 


The  actual  depth  //.  w  hich  may  he  read  on  a  fixed 


\-trtical  scale  at  the  point  in  ijue^tinn,  dit't'er~  fmni  //  hy  a 
certain  (]nantity  '/.  Ihu--  the  last  e(|natiiin  nuix'  he  written  in 
the  form 

(J  _   >;/   II     \-  •I'*'-  • 

A  coiuenient  expre-^sion  whiih  i--  sometimes  used  to  determine 
tile  \iilume  of  tlow  in  wide  rivers.  The  coefficients  ///  and  n 
are  constant  at  the  ^anic  point  for  all  depths,  hut  \  arv  tVoin 
])oint  to  point. 


T.Mii.i    i;iviN(.  nil    \.\i.rr:s  or  »/  wd   /.■.   iin-  vwx  htinc; 

A     M!    I  RI      I  'K      \     n  111  1. 


r,'>»  .ilily. 


AutlMfity. 


|:-1 


M.itii'--.     liiiilKc      on      tlif 

Sf  iiu '1  = 

Roaiinr     tiri(lK<'     on     the 

Loirr iSii  1071) 

<'Minf  bridge  oti  the  Ailda    100-3.2  h.    51)4  5. S  h. 


2.3      CuviiKit 

.S2    (Iraeff 

.0      I.ombardifii 


\'oih  — From  .11  rjtaminalioii  ..(  a  large  number  of  gauginKS  Hres'-e 
infers  that  «  =  .85:'.  gives  better  average  results  than  «  =  .%v,  (Art.  10  and 
p.  a50^>  The  latter,  however,  is  equally  safe  unless  it  is  necessary  to  pro- 
virle  aKain"=t  tloods.     (tnii.  Jft  Pi'iits  tt  Ch^iu^if'fu  I^')7.) 


G.-1UGIN<: 


1  t  "^  "1"  *^  *' 


i 


—  0  f  I  f^  -r  "^  o 
f»  tn  M  ^1  f^  f^  '^ 


ci  «/.  -r  i/>  t"»  -w^ 


-  \  '.  ^.  -. 


I-  t^ 


'Lf.  IT-m    I-  -t  "1  f    >  ' 

^  f^  -r  -  c^  I-  -T  <■• 

r^  O  -C    -C  \n  <r,  \rt  \ri 

5  5  1  §  §  I  g  I 


"^   .     I-  c»  r-  n  f^  *-  O 


C  p»  I  '■'  O 
ro  O  CT  I":  ■*> 
I-  f  -  c^  r^ 

r^  m  "^  N    « 


o  «-  »-  CI  n  r^  «^  T 

,  fj  w  fi  ?i   ri   ri  «  N 

-^  -T  Ct  C     1-^  -    «  ^ 

^.  c)  f*  f^  -r  r-  O  f*^ 

r  t^  w  —  O   J*  f*)  ''  '  ■* 

-I  «  «  w         -   ~^  "" 


r^  (*^  ">  N  «  «  »;;  :«  ^«  *;  "  •  5   -   ^ 

8  8  o  8  S  5  i  5  I  2  3  S  ?  5  5 


/;    -T  I-  w    •-   »n  »^  f  ' 


tr.  C 

3       f-     I-     I  - 


1^  t^  r^  »'  r^  X:  '*>  en 


Z    r 
~  V. 


1 1^  "i 
.  i  2 


woe  c  ^  ■*  :j'  ■£  ' - 

C   1*1/^  c.  r«  ■-    -    -*^ 
tn  w  «  N  CI  CI  y  —  ^ 


X     S     X 


i  3  i  I  I  §  §  i  g  15 


?5S55?i55: 


',  "-.  rn  m  f'.  <•"■  f^.  ^'^  f^  ' 


g.  i. ,/  j 


•too  f<  ••'  " 


»    CI    f^  r^  -T    T  "'■ 
■t  T  t  "f  -T  "T  -t 


o    t-  1^^  i* 


i      I 


>5  'f . 


,   f*-,   CI    •- 


?  T-  -'  ?  I  -  ^1  4  N   -    ?,  2. 
c  ^  -a  "fiT  li'  '^'  •^'  '^  J 

■■  ;   «,  I,  I-  .r.  C   CI  .r.~   >- J  -^  =  «.  ".'  °  ■"" 

; i  o  i  ci  -v i  t^s  ? a  -  ^  :? ^ n' -  - 
.^. :  t  >=  •-  i  ?  5, "  ■_-  i'  ~  ?  r,S  ~1?, P 

?  c  o^  ?  c-  •«  ^  '^  cc  "^  ';  '--  '-■•  '--  b  J  b 

ll'iHl  I  8  5  5  i  8  6  5  3  8  5 


■  C'  »^  C     ^  -    ^'  T  »r.  ^ 

1  f.  -t  -r  t  ""'  "■  ^  ^'  ' 
1  m  ei  c   c»  O  o  'C  f^  ' 


-TsC  r>- 1^  "> 


,  ^n  .r.  .7,  ^  C  O  ^    C  O  O  w  w  ^ 
-    N    U^   C^  t^'iO    -TOO    ^  9  ^  ^ 

"^'  'r  CI  •!:-;;  ^  '^' ;-,  ^  ?',  u  CI  CI  c.  ~>  CI  «  t«  CI  CI  «  « 

S  I  8  ?  i  5  8  2  8  8  «  5  8  ?  i  i  5  §  I  ?  °  8  § 

^8iS68Ss8855Si55  5i5  5-o5 


"-1  fn  -1 


-•x  'O  ■'■    '" 


H  1^  cn  f*  t^  N  c^  1^  -r  O^  •T^-'   "»■  '^ 


O   »n 


H^ 


t-. 


5j>  ,-j  -t  o;  M  c  ci^-^  ?  J  "  P  T 


1 1  §  8  I  §  8  8  S  5  8  8  8  8  6  6  5 


C-  m  .c  r%  C  -t  -  -o  ">  'T' 
^  w.  -f  T  (^  f^  c^'  C*  CI  —  t.Tl. 
~,  I/,  U-,  in  in  in  in  m  •"  "T  ^  VT 

X    X    X    S 


A  X 


7S  s  5  R  r,  r,  r:  i:  ^  ?  .-t  ;=  r:  ic^-fif^i  f: 

^  1^  I-.  c 


in 


.  <»  m  c<  c  ">  -c  /> ; 


5  iC  8  c  ^  «  - '8  8  "■?  5-s  «-i:  ^  f.^  L- '. 


4   en  t*".  «  "  c*  f  •  ^  2   « 


9  C 


t.m  5>  e  - 


(,.■//  >,l\>,. 


p--^ 


U 
< 

5'. 


II 


II 


-     C 


::i  T 


X.   — 


u. 


M  ^O   "tc^W'C*   ^"TC  r^r^«  fOCf*cr  '*'r^r«o  —  w-c 


•  tn  r-  c   t'v  c  -C   f*  "f  r-  in  f-i  ■ 
^  «j  1^  o  «■•  »ri  -r  t^  f*"»  PI  n  —  . 

'.  f,  f,  r^,  f.  p^  fn  t^,  fi  r-i  f^  r*",  c 


-r  -f  »/.o  c 

i^</.  cr. 

a 

n 

-f  -t  t  -r  t 

-f  T  -r 

T 

u 

C  -c   "T  «n  e4 

C    •'  (« 

t^ 

- 

CI  »n  c  o  n 

—    C'X. 

-/I 

rr- 

«    O^OD    O    •'' 

-r  I.  — 

*n  -r  -r  1-  "T 

-t  -r  -1- 

-r 

§§!§! 

^      X,      X. 

6  5  3 

^ 
X 

X 

j^ 

t  /j 

c*  •»: 

tM-.s= 

0 

(*-.£ 

c 

p^ 

c 

c^  « 

n 

r^  0  «  */>  t'*  0    M 

w. 

i* 

-r  T 

N     CI     74 

M 

CI    CI 

00 
Cl 

Cl 

Cl 

C*    CI 

ci 

C  C    C    G    O   -   - 
C4  fi  tn  r^  (^  t^  cn 

(*% 

b 

z* 

1^  t^. 

!■>  1-    J- 

-f  t  T 

r'. 

-^  _ 

CI 

If. 

Cl 

ci 

Cl 

—  1^ 

C    1- 

Cl     — 

U-. 

-f  rr.  -r  tr.fr^    —    If. 

■^ 

8  •? 

8  e 

MJ 

§ 

5    3 

§ 

x 

X 

-.  ^ 

5 

^M^ScS 

5 

o  »/"'  c»  :r  I'. 

ci  '^,4  4  *r. 

oo  JO  c/,  x)  a; 


•.(/l    r-.  I-,  fr  I- 

■  y.  c^  5-  r  r 


^1  Cl  ci 


-/:  C   -t  ri 


f*^  M    Cl    ( 


■   -t   Cl    —     ;   'X-    I 
Cl    Cl    Ci    Cl    —   . 


f^.  r^,  4  ' 

C   =^  C^  ( 


Sc  =  2  X 
5  5  ?  5 


£92 

5  i  ; 


;  ;  R  s  s 

J  ?  5  e  J 


c  c  i 
I  g  J 


.  1  -  r 


Tr-.c    «*.  C    C^—    -Tw 


5  S  i 


C  —  Cl  -t  r-  ; 

!X    I  "•  w    If .  T  • 

-r  t  -f  T  -r  • 


i? 


•  "T  T  w  w.  »/%  I 

»;•  Cl  (T,  -+  C  ^ 

■  T-  ^  ^  ^.  t*,  ( 

X    X    s  X  X  ; 

:  J  S  s  5  1 


5  * 


O  'C   I-.  I- 


f^^-C    O    T  *-•  C^  **^  i/.Oj    C    fi 


n   c*   C4    ri   Cl 


^.  •»  >C  cf. 


•  r-,  CI 


Cl    c  f*"i  i^  ■-  or    « 

Cl C  C  O    2-  ; 

I'-  r"»  r^  —  r-^  r  ^  ^  , 

X  5  X  t-  X  g  ?  : 

X  g  X  X  X  2  V   ■ 

5  §  5  5  5  5  5  : 


X  X  S  g 

X    ^     St    ? 

5  C  £  5 


^  fi'-c^w  r^»ni-c—  Cl  -Tw.r^a.    c—  «  f^»t*n^  i-o 


r--  o   -t  (^  N  O  C" 

*''.  r^,  r^  fO  r*^  (^  Cl 

Cl    C*    Cl  CI    CI  Cl  Cl 

s  g  a  5  §  8  s 

5  8  5  6  5  8  8 


•  ^  C    u^  "1-  rr 

N    Cl    Cl    Ci     Cl 

Cl  CI   M    Cl   ri 


c  2  S 
S  &  5 


£: 


M     Cl 

Cl    CI 


»*^  tn  I 


Cl    W    -T  ""1  ■/" 

ri   Cl    ri    D    n 


Cl   O-  ^<l  O 


-  .  t^,  r~,  t^  1^  I  -  I  -  -t 
in  in  in  m  >r>  ir,   .r,  m  "f  T  *T  T 

esgggsg: 


N   t*>  in 


8^, 

5  5  i 


!■-  C  C   "■  M  e*^  T  '^^  mkC   l^a: 


r4    C4 


y. 


§    §    §    I    I    I    I    §    §    ^   '■     ' 


S  : 


Cl  »r,  TO  ^   in  -/"    * 

Qo  oo  ao  r^  c^  c*  ; 


-  5 

•    "    Ct    Cl    Cl    m  r^  f*".  -^  -t  -t  in  .f.    /. 


c.'ii  (;/\G. 


3'J 


X 

c 


c 
< 

11 


•r  r^  f'.  1^  o  T  1^  -  "■'^  ;«  "^  ^  "?S   "•     : 


— ,  fi  ci  fi  CI  c'  M  N  fi  r<  f<  ;;  f;  "  --' 


S  3  s  ?  ; 


S  5  *  ?  ?  'J  5  5  ;  5 

C  M  -r  c  ■/>  o  f<  -t-  -:;  "-^ 
—    -  --  — -^  j.^^-^-^ 


5  I 


i  t  ^  ^  3  ^i.  3.  s,  „-  £  /c  -f.  -i;  7,  i 


c  5  i 


S  S  ;  5  5 


^ 

/: 

•-^ 

^  t; 

fi 

-f 

r^i 

ffi 

^ 

4 

n- 

t/. 

w, 

C 

" 

*- 

c 

r* 

f 

r^ 

«, 

1- 

- 

»rt 

X 

T 

rt 

(' 

r-v 

"^ 

"T 

-r 

rn 

r^ 

w 

ei 

ft 

fl 

CI 

f» 

fi 

X 

X 

X 

X 

5 

i 

« 

5 

X 

5  5 

J 

5 

■;- 

« 

\i\ 

r-* 

X 

"c 

;^ 

,/■ 

t/^ 

u 

tn 

>/ 

^ 

^ 

tr 

tn 

f 

f 

*n 

en 

f. 

3 

fi 

\r 

I-- 

3 

^ 

o 

0- 

*. 

'/•j 

rr. 

«, 

1^ 

f^ 

1-^ 

I'. 

T"- 

1 

§§ 

1 

5 
6 

1 

Ijj  - 


w 

m 

< 


0    T  t^  C"  r^  -t-'fj    O 


fi  If.  I'  r'  ■ 


3  2 


6  3 


(*-l 

f.  ■■   f»    C    C    t 

r     ci 

^il'i'ii 

ir.  i(. 

T" 

X 

2x25x5 

s  ^ 

1 

5 

§55555 

0  5 

,'f 

l^  C  0   C*   t  »r> 

r«fK 

c 

f4 

ci   H   r*S  r^  rn  t^ 

r',  f-^ 

-t 

*n 

%n  \rnn  xrt  *fi  u . 

If.   u 

Ul 

-  -J      ■_"'■"' 
2l5  _  SS?' 


xyx^xxxx 
gxg35xx,  ? 

—  N  t  »/■•  O  I"-  r"  C 


-  5 

=  «; 

3 


t  !  8 


?,  8 


M  f.  t  »'^  C 


^  ^O  U^  "1  »f>  "^i  t  "T  "T  "T 

T^  rr>  r^  f^  rn  f*^  r^.  ^.  t:;"-  tT' 

222xxx*xx^ 
5  5  55555555 


f  I  o  ''^   r*  m  -T  ci 

-T  -r  ?^.  t^  t^  "■■  '•■ 

r*-,  fn  f*^  »*.  r*~>  f^  i^' 


t'.  r^  r*   «   f*   f<   c* 

m  r^  **^  ^-  ^-  ''I  '^ 


inmiiiii 


4  4:    4  -,    4 

«    C«  N  r* 


r-  5- 

5        5 


1  = 


u 


^■5   S 


«  M  r*  w 


w  w  w  c»  "  c<  ct  c» 


c  C    -  ft 

*f  ■<■,  .r.  .^. 


: :;;  r,.  ■j:;..^.2  2  2  i'--?  ^'5  £5  5"  p.  r-  ?• 


(j.4t  (ilSO. 


t^  C^  t^.^  'Ty   O   •+»A^ 


I  -*>  f  I  f^  \r>  I  ^ 


■1    to    3   -   tl 


t«^  ir.  I-.  C  C    M  "too  O   w^  C    "t  ?■ 

—  r'  I'  K'  "T  «  c  uD  i-»  •/>  -r  f*  * 

f*-i  r»  n  N  m  N  CI  —  —  '-<-«  —  — 

f*  c*  n  ei  ei  ci  f*  n  ft  ft  f*  « 


2  -H  1^ 


ei  ••  C'  w 

o  o  ~  - 


§Ugiig2Sg85§H^g555S8§S 


ir*^  -t 'O  tn    ^  O    W    t*l»  WjO  «    ^  0    f*    «'<  "T  "•  l"*'J5    ^  O    fl    f^  ^ 


t    s 


;-*  r 


^  f«^  i-H  —  tr\  ^  r^'r>  N  I  >  PI  I  -  CI  I 
irt  «ri  -t  T  ^.  ^1  f<  ■•  '-  C  C  C-  C-  I 
i'»i".i-*i*i^r-«-i-»i'>i-i"*i-*J    "^ 


e 

i 


-     A 

""    U^ 

.  .   X 

.    1 

—  r. 

~  >. 

u.  J! 

?:  N 

r  s 

-  < 

Si- 

■-  S    1 

iM 

y.  — 

<    L. 

3-  - 


n 


5  « 


^  r<^ ; 
>  o 


to      o 


•*.  ? 


3-  3-  ^  (y  ^ 


I  ^  O    w     C    w     IT.  W.   •/ . 


??• 


X    X    T,    «    X    B    X 


t-M    ?>  O    -    "    t'' 


«  X  *  ^,  ■ 

X  X  X  « 

5  "^  5  ^ 

»  X  fi  X 


r^  ''^  ft     Pt 


5i< 


ff"»  r*  c  *r.  "f  m  PI 

,  f    f^  r^,  f*^  m  ("^  r*i  "■. 


2  ?'5  '--_ 

(J-  ^  f*%  f*\ ; 


iligS^^sSs;?    fsll    5 


is 


tr.       r*.       c 


5 

I-- 


f  «  ■.    !    :     ft 


•  <  ■ 

.    . 

•  -?  • 

■    5  ■ 

* 

*  • 

X 

} 

-. 

", 

O 

1^ 

Cf) 

?*     • 

s'- 

•  ^= 

■5 
-    «: 

^  , 

•2. 

?■■ 


7 
s  ■ 


l:::* 


y. 


'  i 

?i 

:  r 

7    C 


X 


,  •5- 


,.-//  (W.V(, 


l§ 


■^  £ 


t  " 

I  5 
i  5 


^  f •  ci  r^  t^  m  -r  -f  -t-  -o  •"  .Ajj  ■?  f  f  1;  tr  I^TIT'  T 

~   C-  rio  CO  rn  CO  ao  U3   '^  '^  '*'''"  Lr    ^  ^  ^  ■£  "^  ^  "^  ^ 

,,^  TL.-  ?••.»/>  I  ^  r'  -*  »^<  •'  c  '-  w  '•  f.  >^  •'•  '    o  ^  *^  "t- 

.  •  -i  ^'   e   ^    "   -J   ^  ^  .^  —  —  pi  «  ci  «  PI  n  «^.  f'.  '*!  f'. 
i?ii'.  =T=r^5r=T-r-t-rt-r-T-r-tT-t-r-T-TT-' 


-1  f»-,  !>,  -^  -TOO  ei  1^  f«  r-  M  I  -  CI  «  ''^  ^  ir.  —  r^  • 

c  J  •;  ■;  i  "■'  ■'•  "•  "•  i'  'I  -^  !!•  y  -  i  VO  S"  "  " 


SsS 


5  5^5*5 

u^        r    C    'I   T  wi  I-  U- 


5  S 


-f  iri  in  i*.  »f.  ir>  t'^  J 


_  5  5 

1  -1-  c  e  -  !'.  -r"   ^  * 
5^  «  '£■£•?"    ^  :' 


:i 


-I-  —    r    .jT    CI    £ 

?    »    «   F 


X  X  -.:  ». 


TO     /.    «     I  - 


tj     /J     f.      ft   'O    r<j    TC    _ 


2:9: 

5h 


?■  C    •-    CI    f*>  T  w.  O 


»»co  c-  e 


1/1  iTi  «ri  if^  «fi 


:      5  »■  e  3- ^  3- o>  e^  o  o  5"« 

r-,  ^.  •     *  fi  M   «   CI  fl   Cl  R  w  fi  «   fi 


!*,;*.:    * 


fi'  l§§l§lll§§HI'  HI 


«*. 

— 

:< 

^ 

^ 

•t 

Ua 

U^ 

u. 

J- 

r. 

H 

y. 

^; 

«: 


f. 


^ 

>< 


1 


i* 


< 


f^  r^  ••.  i*t  •*.  »*  •n  1^.  •*%  •♦  •*.  I*.  I*. ' 


ii*    i   I  ^  V'*'?8  ??f  ?  s  -  -  - -'i'!""*!"*'^**'** 


C.lfOIMj. 


\     O^^f^t^C-fttCf 


f'.  lA,  r^cn 


•Si    f 


y. 


te'.4 


J  J. 


2C 

V 

U 

a. 

X 

3 

V. 

y; 

N 

< 

^. 

s 

,^ 

M 

A 

t- 

< 

•« 

<  - 


•  -t  »''iO   O   •-   N   f ,  f^  r^  —   f*  ^r^    O    -t 

i/i  -r  r*-.  r^  N    ««    C     C'  '/■-  -r,    r->  -^  \C    wi 

I  u".  »r»  Wi  >/i  Wi  iri  ir.  T  T  1"  "t  -t  t  "t" 


-f  m  * 


§§§il§§§§s§§ii 


r'.  -t  «•-  l-OD    C  C    —    f*".  "T  tf.  O    t^oo 


6 


f,  t^  n  -t  -t-  • 


(         1 
I  u     rs 


2  S 
5  5 


?i: 


I  %• 


C  r^  T  CI  CO  (*■■  —  r)  C  r*^  —  r"  I ' 
■■  C  o  C  C^CCC^J  c/;v  n  i-i" 
ir  u^n.i.-f-t-r-T-t"t-rT-r-* 

"5*  *?'  '  "t'  ?'  J-   j*  ' 


5  5 

X     X 


t  —        1^ 

1/1  .r.         -f 


Z  I  -  'J. 


3-  s 


c  ?■ 

S  R- 
5  5 


c  :    e  : 


{•    5-   • 


5 


fi  = 


'  •  5" 

X 

f«^ 

fl 

r* 

fi 

'  J_ 

8 

ft  *     ' 

:    :   : 

3 

X 

:§:: 

:   :   : 

:  o :   : 


UJ 

fc 

\J 

^'t 

< 

Ji 

> 

u 

■/.ii 

■" 

.  «• 

5:  fi 


X 


?s* 


!  I 

V 

i  a 


1^   n   0^-0  M  {7^  >#'  N  '«)  «n  H  «  in  * 


|.A- 


O   f*    "t 


GAiCjlNC.  3*7 

I  s  5  -  «  -  -  •?.  ?  ^>  3.  s  ?:  2  •&  j  f;  ^ '^  2 :- - 


1 

X 

> 

1 

c 

Y 

tu 

*": 

< 

u 

03  g 

CD 

l«i 

t! 

t 

!— 

> 

.   s 

into   —   ^'*.'   •-    "T 


jj-        -.#14 •i*tf*^e^fnfnf^^ 


?  / 


T 


^t^ 


lI» 


3»« 


u 


0 


?: 


■<: 

,. 

II 

C: 

'.^ 

5 

S? 

,~ 

V 

_j 

1 

^^ 

»'. 

— • 

a 

^ 

u. 

•y, 

Ui 

-M 

ai 

1- 

/, 

>-■. 

y. 

^ 

u. 


U 


f;.-//  o'/vG. 


«  c   —  ■/!   -f  : 

■^  -t  "'  w^i 

1^  o-  r  r 


C    «    tf      C*  M    ir,   1^   0^  «    -t^ 


1^.  r^,  r^,  f  ,  f*l  d  r-i  r*-j  rn  rn 


"■ 

—    ""    " 

Z  Z 

X 

"t  - 

I'.  I'. 

i  8 

X     X 

■i 

l-«   c 

2   - 

fl  • 

?■  ri  n 

c  o  c 

tr, 
«  - 

X    ' 

T  f. 

S  2- 

in  \n 
t*    CI 

5  5 

M 

-r 

.o 

5.' 

c  '    ' 

.'    6 

i  i 


1  ;,.-^  :        r 


< 


£ 


s 

,*2 

? 

f. 

% 

? 

s 

s 

fi  .f 

r^ 

T  »r.^ 

1  « 

c» 

*#» 

£■ 

is 

0 

U". 

S 

1^ 

S       ': 

■  "T  -t  *A  "I  W'  m^ 


(,,//  (,7\(;. 


.^«'> 


IN     HA/IN  ^     M.W    roKM'I.A.  /'.■ ' 


'(-;„) 


OK      I  ill;    MX    '   I.A->I>    I      1"    VI.    1'     25". 
,;,   i)K   :    -    .t  w/,    W  IIF.KI 

Ai(  iiKHlN'.       x--      I  II  1 


INl  1     IS    A     Mi:i  Kl.    "K    A     MX. I. 


(    T.^S    III. 

ihf  I'nil  bting  a 

.Mfii.v                Fool. 

^        ,        A     1     . 

om!  »8.4     ■•^'°^\ 

51    1 

ojil     y3.2       .'"O"'* 

M'i 

"t'5    31   7        "^ 

56. i 

.o3li»     i;               ■  "=         ''4    i 

,..n4                            '^  • 

.■<»'>■< 

«.».7 

.«>74' 


''I 


77  7 

T    -^ 

\\^.'\ 

81.7 

8]^ 

84  » 

84  V 
»VT 
8t>  J 
t>6  7 

ft7.« 
i>;  8 
8H  i 
ku.i 

•(O  V 
•Jo-7 
VI.  tf 
91. S 

gj.l. 
Ml   I 


■4  i|, 


13^ 


t::A 


320 


VAUKS 


0.41  V,7.Vf, 


)I     />    AN'I) 


!-()R    111!. 


-IX    CLASSES    I   Ti)  \I.    I' 


25'" 


IN    l',A/.IN'> 


NKW     lOKMll.A.   /■. 


v/.  OK  7' 


■i'mi 


U  Ill'.KI. 


f, 


<■   1 


y 

\  m 


(IK 


l.'iT.'l  S  "II. 


Al  t'()KI>INt>     An 


1  1 1 1; 


l.'Nir    !S    A 


MtlRK    OR    A    iO(,>l. 


V.ilue  of  m, 
till-  Unit 


Cla-s  I. 

the  1*1111  bfiii^' 


fi  ».- 


II. 


Mttri- 


Koot., 


Mclrt. 


ihe  L'liil  htinuii 


M.tre. 


Cms  III. 

Hit?  I'nit  hctiii;  .1 


F....1. 


.1. 


i.ro 

1,70 
l.lv 

t  » 

34" 
a.fm 


l.r» 

4.0. 
4.5" 
S-on 

6.V 

7.ur. 
7.V 

M.JO 
l<»   O) 

1 1  .on 
li.oo 
I  l.t-O 

14. OU 

l>i  no!  5>.4« 

17. .Ill  »n.77' 

18  00!  KQ.fitii 

vt  "A  fa. Ml 

.•ri.ilD     t'i   6«j 


3.131     .. 

3...S 

(.*! 
.S..I51    . 

^  14 
<i.57 

7.»'l 
7.8?1  . 
8  mi 

S..84 
lo.$r> 
■  t.nl 
il.8l{ 
1J.47 

I ).13l 
14.76] 

lf..4  I 
18.04! 

'1   .13 

'\f»\ 
'7.*9, 

3>.'7 

.181 

V,.J7| 

4t.n<i 
4S  "Hi 

4<>  a> 


Bi  o 
8  J.  a 
81.4 

1  8j.6 
||,.8 

!i|.o 
i  8,.l 

,  Si. 7 

;  8.i.i 

8)  4 
8|.t> 

«J.7 

8)  8 
8l.'( 

:  84... 
8,.l 

1'4   ' 
,  l<4.i 

84  7 

84. 8 

!    84.1) 

85.0 

i    85    > 
85.. 

rl     " 


8V4 

8V5 

8s.6 
8l.» 


8s. a 


0037a 


,oo:ftq 


U0366 


l-uoV' 


OOJJ6 


148.4 

!4*t.<» 
M'-l 
I4<-' 

|.-o.    . 

15>'.4l 
ISO  5 
150.7 

Hl.l 

|J«.4 
151. f> 

m.8 

15a. o 
l>a.a 
isa.4 
1^2. (i 

na  7 
IVJ  n 

'.■;j  .< 

I5<.4 

■  •jl 
1S4  o 

1:4  a 
154.3 


.<iiji 
.iiiji 


74  7 
75.0 
7i  4 
7-.'> 
76  J 


.,..41.8    115.1 
■  OU4u5i    1  ts.8 

1 ''.5 

..x»4oa    1 17  4 
•  O0J9.)    1.18 


oi'8, 

Ol^^i 


"54-7 
155.0 


76  9 
77.» 

.o'»v    77.5 

77  7 
77.') 
78.1 
7I.S 

78  8 
7.<.l 
7J  4 
7,.6 

7>.8 

Ho.u 

8... a 
.olai:  8u.4 


.01 ;'." 


.01  >8 


.uli? 


,ota(' 


IJ5 


8U.5 

I  80.4 

<i  81. 1 

,    81.4 

I  81.'. 

.olaa    81.8 

8a  o 

8>  a 

8a.  3  ' 

8a.4 

8a. 6 

8».7  j 

8a. 8 

8}o 

«,.i 

8M 

81  4 

»|.J 

Ki.4 

8).7  ! 

8«.8  I 

»V»  1 
84.0  { 


.ootgA 

.oojyj 
.or_^«i 
.00.J87I 
.onl84' 

.00381 1 

.\     I 

.00)78 

.00375 

■ooj7-a 


o.jN, 


.003^6 


".i*'' 


IJI1.3 

■W.«| 
U0.3I 

14...7J 

.4.  5| 
14!. »' 
14/. g. 
141.1 
141. 'j' 
144. i 
144  ''i 
141  <> 
14'i.  1 
liS.** 
14  .^ 
U^.j 
147.-1 
lift.'. 
147  8 
148.' 
148.5 

14"   <» 

■  4      t; 

■4V  *' 
i4«.8; 
150..., 
150.4 
150.5 
15''.">l 
151.1! 
■5'  li 
15'   4, 

"'5' 

I5»  ' 
I5». 


,ol'.u 

..•St. 

.ul>', 
.'•'54; 
.01^  ., 


L.i4li] 
'4' 

•  14 


5»  " 
t"  5 
61. J 
<■>  o 
6a. 6 
'3.3 
6)  H 

"4  1 
«4.8 
65.3 

<«  4 

«7.i 
f-l  7 
'K.a 
'S.7 


Mil  '■..'■ 


..141 

7".? 

"14  . 

7'. 5 

Ill  111 
01  |K 

7?.i 

7'.7   j 

...117 
.t.ltf. 

71-'  ' 

7.1.7 

...!<« 

74.1 

■".'.'* 

74-  = 
748 

■".'." 

75-'    i 
75.4  ' 

.01  )a 

75.7   ! 

.01 ,1 

7*  « 
-ft  4 

■".'.* 

7r..» 
77  • 

01  ao 

77-4   ! 

.31  aB 

77.7   1 
7B.U  j 

79    1^   t 

..X15011 

•  ■■0447 
.H.4.JI 

..048* 

.  oi8a 
'•'47  J 

.  .M  47ft 

..ftJ47li 
....466] 
.•-4"1 
.*04'.*)I 
.U04-4 
.1K1451 
..x>448 
.OU445 
..<i44a 

.'  »>4  W 

.u(J4^^ 

.•<..4to 
.i<»4a7' 
.no4a4 
.>li4..i 
<Ki4t8 
.»r>4t^ 
.oi.'4l/ 
.U04u8| 

.00405' 

.UQ4ua| 
.<»3gu 

.001./.I 

.ooWli 

..:;,.! 


ia9.e> 
ill.o 
lla.4 
■'J  4 
114.5 
115.6 
116.5 
117.4 
118  I 
118. g 
120.  \ 
lal  5 
i»a.6 
i'ii 
114.4 
135. J 
I  j'i.'» 
lali  8 
117.- 
la8  I 
ia«  5 
1  ;o.6 
'11  7 
I  tl.ft 
'Jl  ^ 
'34.-- 
I  '.t4..< 
■J.5.5 
Hft  .. 
136.. 
1.17..1 
•  .17.4 
1,8  4 
I  iM.i 
1  :l»  7 
140.  a 
14U  7 
141. a 
141.8 
14..    • 


IS- 


Cj.-il  ulM> 


rsn    IS    A    MMKl-.    ..K    A    KH.l. 


the  t   lilt 
lulli^  •' 


ClA         IV. 

,,„.  r."'  ii.Miii,'  .1 

Mttrr  1        ' 


C.—s  V. 
lie  I'nit  brini: 


Mrirr 


I    l.A-.     VI. 

■,lic  Vnil  beine  a 

Met"'.  ' 


M,ite-,K"nl-       i 


p^ 


ll-t 


32- 


c.in.ist, 


VAI.l'i;S    ()|-    /■    ANM    ,      |iiK     1111;    SIX    CI.ASSl.S    1     |()   \I.    I'.  'J.Mi, 
IN    I'.A/.lVs    MW   |MR\iri.\.   /-;■         />ii\   oK   -        .  *  w/.  \V  I  i  Kk !: 


Jl     -       ■*'    ^    .  -   HT    »    l-il.    I)R 


!.'>7  il  \  mi.     A(  CiiRDlNi.      AS      1  111. 
INI  1     IS    \    mi:  I  Ki;   OR     \    Idol. 


I.tx: 


P'^ 


the  I 'nit 


Mi-irr    !■, 


I  ..ASS    IV. 

the  riift  bvint;  a 


1 .  .'i 

4.  11 

1  no 

5-»5 

I   70 

«   e8 

T    So 

?  •<< 

i.8o 

»5'.47 

.ul' 

I 

4  «o 

I         :   ■ 

-    '•  1 

♦  ■sn 

-,   •  *> 

^      '1 

.01  s 
.f.l  ^ 

-.00 

«./ 

-  '^    . 

7.W 

.1.(1 

1 

.■■U4tiO 

Irf. 

»  ■-> 

.'".2 

, 

.■.>.4'7) 

Iji 

8  J.* 

i7.K.i 

•    1 1  < 

'     .4 

.i«i4f4: 

I.. 

.,  iij 

.■<.  =  3 

..MS 

f.7« 

''J4';f 

1.-. 

»      ' 

1.17 

...l47 

'S.a 

.'■448 

1.;    . 

1   .v> 

i».ai 

.014' 

.8.5 

■•>M4< 

11,   . 

1  .  Hi 

i^.-j 

.OI4) 

'-(■I 

.,,438 

li~.  t 

1     .Ul 

i,..r 

KI4  1 

'  t.'i 

...4,1, 

l.**.^. 

.  .14. 

-  ■  4 

..«J4(i 

*»7 

.f.I4I 

:"•» 

.104,0 

1.. 

1      ' 

.|.I4<) 

■•"  f 

00417 

1  /  . 

'■■■ '» 

■■  4  J 

.01  1-, 

7'.7 

.uo4»4 

11 J   r 

7.-W 

^i.?' 

•• 

.»?   1 

••       1 

n.n 

■ 

- ,  ^ 

...I   * 

m8 

(1    \s,    V, 

■lie  t.'nil  bcin^  a 
M..-ti.-. 


^...=;| 

.'»|8| 

4'i.'> 

.ooAf.!' 

8|.T| 

.0/7.1 

J. 13, 

.0215' 

4fi  5 

.00654' 

84    2 

.oi''7 

i-^l 

.•^.JIJ 

47.0 

.-N.i 

8.i..| 

3.-,. 

.••I.  K 

4^.-. 

.>,'„! 

f,..; 

-,..,4' 

■''■■    1; 

4*^  ..J 

..'.-/ 

k,i    '. 

!   .  \  1, 

The  I'llit  hfint;  a 

tri-.  Kiitit 


..mRvQ 

66.5 

.r»j8i| 

67.6 

:« 

6S.S 

•so 

7"- 3 

...5 

71.9 

-  741J 

7t  5 

.0.714 

74-'* 

.     731 

45j 

.■•^:4 

Bl    - 

...it*.. 

4"'  4 

.  V  irj(i  i;  g 

1-4. 0 

..>?I3 

4'^    1 

,.>^4r. 

t;.7 

..,.4, 

4    ... 

.    .208 

4"' 

M.*    14 

sr.i 

....■4-. 

4".7 

..a..4 

4«...> 

.rx)t.M' 

y*.'. 

.'318 

43   5 

.030I 

4  J  7 

.C-*'^!  I 

S<J.9 

"3tl 

41   .1 

...IQ-4 

5'>-4 

.•jiyv7 

.41.3 

.ua.-7 

44... 

.   ../. 

51.0 

.■-.3.481 

<)3.1 

.(..334 

44.«. 

'  '4 

^1.0 

....5.U' 

93-! 

.<<a3i 

45.> 

•  o..'.:4 

'  ti 

5.'. 2 

Ori^Ss 

04.6 

.tr.Jt:.' 

4<.8   ' 

..»*(5 

«4H<i 


-ul4'* 
.0148 


8i  ■ 


^  o 


113. ? 

I!  T> 


J;  « 


X 

< 


GArcisc. 


3»3 


r.  i?5,L"r  S-'i;?  2    f-"  c- '^?>???^^ 


\r\  t^  S.  '/-    C> 


w     —     —     Mf»  ■-     —     —     ^     —     ■-•-"" 

iX  —  -        :i.  -r  u-i  c  c   '-3^^   '-    "^  - 
r  *#:    C  r*   «^  "■    *^  ■■   C    ^        T" 


£  r  u .?.  7  '^ ?  2f.    ? .'  'i  ?  r,  ^. :r.3 ': 


U,'^    t  '^    ^     -      - 


I-  ^   ri  <<1  .<  '«.    r   -  "   T       ir.  i«  ^ 

;.     —     .-     —     —     ?l     CI     CI     CI 


f.  I/.  I  -  '/. 


T  w .  1  .  I  -  -/- 


/,  -  f,  U-.  r   C   r>  ;t  I- 


:;i''j.2 f.  r,.;r  ^ ?.     -S  S-;  >■?« "s..*  %  f. 


-  M  t«)  m  !«■ 


t  I  .  a 


u 


3-'4 


gauCmIn  :i. 


^  li^'T,  r-  -r  tr,  c   fn  r-  o        !2  "^  "^  ^^  T  "^  3  ^  -   2" 
^f^  _^   1^-/;   ^  ;   «  -I  r*  -'        -T  w.  '"  I-  r-  :t  ^  ;3  —   — 


»  tn  -O    tri  ^  >n  ,c    fO  "<"    '  - 
*  O    I  -^  'J^    ^  -    -    ^*    ''•    " 


-  r-  r-^  r-  z 


^  I 


1  5 

9 

5 


w    r~i  _-    ir.  PI   ir,  T    »r.   T    ^  ri    '-"i  '^  —  -^r,    «    —    I  '  fl  r^. 

If ,  O   t^'^J   ^  C    «  N  '^  "i  T  w.  ir.  r^  r^  r*  _    ,   —  e* 

,ri  ir^  -I   r^  —  -r    ir.  C  fl  r^  f^  C*  ^  -?*  (*^  O  C    ^  f<  '_ 

„   ,. ,  -I-  -r  ./,  ■,     c    ^    '^  I  '  M  fi   ?i   fn  -f-  ^/>  W)  ly.  ^  ^ 


CI    r'.  -r  -t  "•  »^  O    r^  "^  ^         CI    «    p»    f^  -t  »r>  "^  u-  c    t^ 


-t  1^1  1^^     r    C    CI   M   <*>  in        rl  T  -*■—    1^  "^  Z    -    -    tn 


TSK 


|::4 


■^E 


x. 


u 


-f  -I   c  <f.  O  'O  r^  —  w^  en       c  w-  ^'^  '^  ^1  O  1^  "r  ^   '1' 
M  (A  ?i  T  Ir.  .r.  c   I-  r-'«        ci  «  N  f.  T  ■/!  ^o  ^c  t^ 


^  U-.  ij  i~ CO  i)  ci  c  -t ■-"        c,  -r  -rc  I-  r  :   -  CI  -T 


n  oy  t-^  r*-i  O'tn  ^  ^r  C'  O      oocnmt^—  cct-o*- 
S  w  K-??"  ^c  «-»=•  5>      -  c<  n  n  "t".  mo  t-io 


•^    J! 


C-  »^>    •*    t     '        '       -      '      •  -   'i 


\  -^  If .  -C  ^    t^  O' 


CO         W'C^CCOSx** 


tl    r*.  in  1^1 


CI   tn  U-.  w 


iio-  =  w 


G/tUGlNG. 


325 


y. 


< 


'^,  c  c-  o  <*^  f»  f'  ^  tl  5 


^  S  f7?>".^^-ii^">      -  -  "  "  « '-l■^'^••'"»■ 


;^^?~icn':^??^5     r??!^-:^iSrt?i^ 


as?.s^^^=.-3.   ^-sas?^^-^:;?? 


3 


-  c  c  -1-  o  "I'o  "  >;-     :?  IT  n  n  '.  ?:  ?^  ^  5  "< 

D    t<    f  <^  T  -r  T  "'  "^         _    -    ri    rt    .. 


Sf.  'ii  ZZ  fl^-5  ':?  =       ?  S  -  ^  ?5  •'  -  "  ^ 


fr  ?.  ?  :;ii  fi^'  ^'S  :r    "  '■  -  -  -5  firms'  3 


-*  Cr  r*"i  O    f'*  —   f  ***        ^' 
-   _  ct  M  **-.  f  -r  w^'  u^  *- 


-t  ""-  f  •"  ''"• 


CI  r^  r^  r-"ri 


"  r-  fn  O   f^'  '-  »'^ 
CI    N    r^  -r  -t  T  »« 


,  -r  -  I-  ^  -^  -  f  1  " 


,.    ^.  -t  -  w.  .^  /-    O-  £    N  N    "    r^.  -T  -r  C    .^  /      -    - 

^    .^   fi     -     f*-,   —    wt  —    N 

■I  Z. ^  ^  .7  r.        -  -  -  fi  ft  t^  -r  -r  »^w; 


^r  r   ''  '^ 


5 

ri   -    0   i-w    ft   -x 
CI    en  -T  T  •'■  «^  - 

V        1 

^     1 

2  •- ;;  f.  -  5-  .^ 

E 

1 

}L 

X 

r-.o   C-S    £ 


—    T 

2  ?.£.?.???  8  ?     ?  £.  F.  S.  rr.  ?  ?  S  6  8 


\ 


Seo"  -  « 


o£o-  =  •• 


\i<> 


0/1  u<:  INC. 


C    \ri  '^f.    f*^  ''^  I    "^ 


—  ■»     oN^**^'»*'^**^'^*r"' 


'^;^.T;-.iT«-.i3  r-i-      Ze;ci^^-rw^.r.ci 


W    ri    ft    r*-.  f^,  r^. 


—   —   rt  f  I   r*   f^  f*".  t 


y. 


,  M    O  C     ri 


M    CI    <'.  M-i  T  "  -  W''  C    1- 


-♦  r    -    -T 


"^  '1    z   f'    —   i/i  ^  f  <  -C    "* 
„    -,    —    C^    f»    ri    t^.  r".   -t 


ir^ 


c«   m  fn  -f  *r.  C  O   '*  J 


,  -*%  -r  If  I  c  c  *  "• 


r    ri   i/.-r    -   .r.  c    ".  r-  n 


f*-i  r^  -T  't-  C    C    I  -  '. 


t^  c  "  r  -*  I-  f 


M.     _     ^     C4     t»     ^.   ••■.    ^. 


D    X 


c  *ri   :^  -T  'T* 


':\   t'-\  ".x    "Z  f-    t*         m  C    "'>•-'  -^    ^  —    fj 


<f*   r-,  -T  »r.  ^    C    t  ^  r 


fi   f.  r^  -r  »/■>  C    I  '  "X 


ii*: 


1^  -  -r  I-  r 


_    _    Ci    ri    f*-.  r^.  ".   T 


S    1     -c  p»  r^  "t-  «  '*''  f^  *■• 


f^'T)   in  C*  t  «'"-  C    T*. 

ao    O  *-    —    Ct    ft    r'.  -T   "J".  C 


~ci^C    ^c^C»fiC    'r.O         l^C    -fC 
"    —  —  ..  c*  CI  "■-  -t  -r  t*^'  _  —   « 


fi 


>tK 


K  l!^4 


«    M    («^T»r.  ^    ^^  *- 


:  in  r  "    f»   '^  ■ 


z 


O 
> 


i-t    C^sO    C-O    ^ 


'=^^.2.:5S83     TS8?,S.?g88  8 


M  r^  \r>  u- 


SCO-  =  « 


ot'O' 


d.ll  t.i\C. 


327 


MANNING' 


VAiri'.S    "1 


1  !l 


K    K.KMVLA    :■  =  <i'"/'-     ""•- 


;•;,,     uriN.;    A    MMK.     -K    A    1"'»1- 


Otll 

suii-iii'  ii"l 

vti) 

R,,uk1i  ^"' 

.  ti-. 

Vcrv  >nu 

Smooili  Surl.ici-. 

Smoot 

1. 

Value  ■ 

.f  ).(. 

M111..K 

-      - 

Metre 

y\.,::. 

Mmi'- 

I. 

Mflrc 

K..0I. 

Mctii . 

, _ 

1 

iii't. 

1'> 

61 

I  10 

47 

'!? 

36 

f'5 

?4 
62 

.of 
.  10 
.20 
•3" 

.33 

.lltl 

ii« 
^2 

133 
13* 
14* 

f'3 
1  >i 

'»4 
ICJ7 

114 

1  -  ^ 

4'^ 
45 
4- 

-1 

'14 

41 
4  = 

74 

■  ?■<■> 
I  00 

3.2- 

l.in 

I-I 

t,i) 

•.3'» 

1   if> 

(ji> 

i"7 
I  I'l 

ill 
5(1 

ll>[ 

2.0<) 

i).5'i 

1 1.: 

-03 

■»ti 

1   »t, 

it'" 

71 

12^ 

■  0      , 

ll)l» 

5 .00 

9.-4 
II, .4.) 

121) 

13> 

217 
237 

nil 

1-3 

c)2 

13'» 
"'7 

"5 
7'> 

I IS 

143 

15. 'JU 

u,.20 

i  ■"  l 

2-4 

1  Ji 

Vciv  I" 

—    -  — 

L-ul.ir 

Su'( 

1. 1-  in 

Irii'nul-'f 

MjrI.K 

^uitaie. 

V;.la 

e  (,f  "f 

surf.ici-  IT 

V,unh 

tir.iV' 

,y  ^''ll- 

Mtirc 

.05 
.10 
.20 
.30 
.so 
I.  CO 


.If) 
-33 

.•)$ 
1.64 

3.2s 

2.<)0        O.ffi 

3.00  '1   9- '4 

5.00  ;i<i.4u 

15.00  149-2" 


.Mitrc. 
24 

31 

33 
3'' 
4" 
4; 
4^ 
52 
63 


K.."t. 

43 
4'» 


87 
94 

H4 


Metre. 


21 1 

■J< 

23 

\- 

25 

4- 

•J  7 

4', 

-,.  1 

r4 

33 

37 

4'J 

t)0 

44 

79 

52 

94 

Mi;tc. 


17 

I'l 


32 
34 

37 
45 


Mii' 


31 

34 
4" 
42 
45 
53 
5'" 
(12 

''7 
3l 


15 

17 


'J 

34 

25 

4'> 
45 

2S 

5' 

30 

33 
C'3 

54 

llO 

:'4 

y 


J'» 


hXAMPLtS. 


VK' 


!»:$ 


y 


EXAMPLES. 

,  What  fall  mu'^t  be  Rivn.  i  ■  .  >  .."-n  2'a>j  ft.  fonu.  7  ft-  wide  at  ilic 
I,,,,  xh.  wi.lr  at  tlH-  hotlu.n.  li  ft.  -k-fp.  an.l  .  unv.-ym(i  4"  <•"•  «•  ">• 
water  i»r  sfcoiirl?     (/  =  ,,.)  ,       ,      .  J/, 

2  Drtriminc  tin-  f.iU  o(  a  <  anal  Moo  ft.  l<.n«,  ..f  .'  ft.  i  -wcr.  8  ft. 
„,.,„.r  l.r.-a.ltl..  an.l  4  ft.  .U--p.  win.  h  ,s  to  .  o.uvy  70  .u.  It.  ,>f  watrr  p.-r 
"      ,  -       ,  ,0  ,  Alls.  I  111  1088.4. 

second  .-      I  /  --  ."<)b.l  ,  ^      , 

^  |-,,i  a  .h<taiicc  ..f  30.J  It.  a  l.rnok  witli  a  iiu-an  water  prnmctir  of 
40  !•  I..-*  a  f.ill  of  9.<'  i"i>-.  '1'^'  •'"•■'  "'  the  upi- r  transverse  prolilc  IS 
-osn    ft     that  ..f  the  l..w.T»K>  sq.ft.      Thid  the  .lis- haru-r.     { /  ^  .00&.) 

•      '■  Ans-  yi'-y'- 11:  ft-  I"'""  «^'''- 

4  1„  a  h..r,/o<,tal  t.e.uh  5  It  hroad  an.l  Soo  ft.  Iomk  it  is  .l.sire.l  to 
.a.fv  Mir  w  .u.  ft.  .lusrharK--  an-i  to  let  it  flow  iti  at  a  depth  ..I  ;!  f..  ; 
wl.at  niu-t  he  th.'  depth  at  the  .  iid  of  the  runal  :"     (J  -  .ooh.) 

./«>.  11. 

-  W  ilei  111  •  s  aloi.y  an  o|H-n  i iiannel  \Z  ft.  wide  and  4  ft-  "ee|>.  at 
the'rate  of  •  f..  per  seeond.  What  is  tlif  fad.-  .\  dam  M  ft.  by  y  f. 
l,,d,  .s  fornvd  a.  ross  the  channel  .  how  hi«l.  wdl  the  water  rise  over  the 

.-rest  nf  the  dam  }  ■  '">•  '  "'  4«0'  /  ''^"'«  f  =   '-^'"^  "■, , 

6  A  stream  is  reetannnlar  in  section.  1:  ft.  wide.  4  ft.  deep,  and  l.dls 
,  in  100  neterminell.edischar-M,..thanairi.rri.aetor;  (2)  without 
:,ir.penmeter.     ( /"  -  .ooS.)  /,«.(.)<  46  ru.  ft.  ,H.'r  sec. 

'  '  (2)  ('(15.0S8  (11.  ft.  |HT  sec. 

7  A  canal  20  fl.  wide  at  the  hmtoni  ..nd  havinu  side  slopes  of  i}  to 
,  h-,s  S  ft.  of  w.itcr  in  11 :  find  the  hydranlu  me.in  d.pih. 

,tns.  5.103  ft. 
S    The   w.iter   in  u   semicnnlar   channel   of   10  It.  r.ulins  when  full 
n..wsw,.havel.K,.vot.'  ft.   J..r   ^. 1:   -hef-dl   i>  ■  in  400.      Fi^'|'l  ""■ 

codlicient  o(  trKlioii.  '   "*     " 

,,  Calcnlalcth.  Ilow  per  iia'  no-  ...  lo.s  „  ^o,u  s.a.on  of  a  rectan- 
j-ular  canal  :o  ft.  d..p.  45  't  wide,  the  slope  ol  the  hcd  l.ciug  32  ms.  per 
mile  and  the  co,  lliCH-nt  o|  iri.  tion  pe.  Mpia.e  |o,,i     .  .oo,\ 

,0    Whv  does  the  water  of  a  river  ri n  the  lormation  of  tlu'  .. .  } 

I  ,     Fmd  tlic  depth  and  wi.ltl,  ot  a  re<  tanuuhr  stream  ..f  .M>  s.,.  (t. 
,  ,,..„.d  ,,rea,  so  th,.t  the  llow  ini«hl  he  a  ll.aJtimiin.  ;  also  Itnd  the  flow. 

/  heintt  .oo«  and  the  s1o|h-  11  ins,  p.r  mde. 

,/,„.  ji.ji  ft,  ;  43.42  ft.  ;  4S>'>  •■"•  f'-  l""^  >"'" 

II  The  section  .<'  an  tt<|iieduct   ,s  a  t..i|H'.ium  with  a  l...tiom  width 

,.r  (.  56  f...  a  top  w.n.-.l  of  7.Uf'  <-•■  "'"•  '«  '>«P''"  "'  "'*"■»  "  •  *';^  ''^"'""  " 
6  per  1000.  and  the  (ttce»  of  the  aiineduct  are  of  brickwork.     Determine 


t:.\.-iM/'i  i:s. 


3-"> 


--'s'^»"S,!::'n,»;:;'::;",;:-;;n:;er;«iu: 


4' 


/«,.  (..)  47 '-276  ;  (/')  494.54'<4  :  "  >  4«7.''973- 


I    V 


AnaqiicluclMf  r.-ctani;ul.irsf.t.on 


IS  t.>  .oiufV  V504  (Ini)).!  ^,ll- 


,      '"f"w.,c;ucr  hour  ..l  Un-  maximum  v.I.k.v  •■(  tU.     Assa.ni,>«  as 

c)i  11  i^.  ,  ....i..!   iv    1  ft    t)cr  secoml  in  ii 

Wh.a   ho.i.l    ,s  r.M,mr.-.l   t.,  u've  a  ^"■'"' ;>,•*'•  '*;,.„,,.   .„„j-. 


14 
M-mir 


.     „     f    i  Irnirlli  Mf  I..1  K..<  liL-  Canal  in  roik  has  a  b..H..,n 

\  ■   ,       Tl  ,.    „K-M1    v.-lncitv  of   flow,  vvlun    lllL-    waUT 

^'•"  -'^  ^"'  S:is'"u     i    vrJ^^^-     r.n..;i„..l.,K-..sui.aM.  vaUu- 

lr:,,-';l■!:;;v:l^^:.Us...,.......n.hel^.h..^ 


r    .1        I   ,    U,.,iie   I  anal   i"   <arlhw.>ik   has    il^   sides 

''■■•^^'■'a;;:i.;s?h.i;;..n::u.::...3.n.  wh..nu,....e,u, ....... 

„,e,li..har^.-   .^  at  th-  ra,.-  of   .05    hire*  per  s.-.-.n-l. 


v;,,p,'(l  at  45' 
•A.itir  is  o.:    tn 


,,.,.,  „  0.=    -n.   ";--:■>,.;.,.,  ,,„  ,„..  ,.,,md.-nt  /■  b.in«  S..U..  t.,1 

'''•'•TrV.Jc;    'aI^        :'w   t;.at.  a..or,i.u«  .0   naz,n's  f ,a.a.  the 

"""  ie    .Hi    .!..•  t.o..on,   l-.lamcnl  v.-Iocitics  a..-  .«,-.  .n.  and 

■''"-■wr,'.S...,nc..  ..,n..K.,rfc.,l  cl,...n,-l.  i  fl,  v,  i.lc  .„  l..,.....l 

,,,■;X■;;;;::;;";^..eM,i,,.,...--^--;-;;";::•;:;-;;;: 

second  who.,  its  dc-pth  IS  «.  n.  ^  _        ^  ^^.^^^^^       (^_ 


(  a  slnsiii.  4   fl- 
,(  till-  surface 


,rf.  CaU-.datc  Ih..  fl.w  a.r..ss  the  vertical  s«t.o  , 
,  .«f,    wide    It   top   (<   ft.  wii-   at   I.  fo.U.  the   sloi.e 

deep,  18  ft.  «"1«   ■'        !•  ,„,    ,,o,93r«.c...  It.  |K-.  s... 

U.,„^.8.n.  p.r.n,i.-      (/-.oo  ,  ,,  .,„,.,r  ,rai.e/.o,dal  s.-.  t.on.  has  a 

j:j;:r:jr.;i:r^;';;;,;.f^M.;^^. --.■•' '■ 

" ■'''■■■'■^::-;,;:;;;r:r:::,r:p.....i« -  ^-;^. 

■:::i;",'i„'  .„",;:. ..-  ..-„., .,  v..,u,.„ »« »..."--,,^^ „ 

c:'"- '■- ;;r;:-;;":i'r-:":-;j";:t:.. 

.,   I,.,  liiven  to  the  t.ansvcrsc  prohle  o(  a 

21.  What    o  iia  111 


|l 


.>.?° 


I  x.-i.ui-i.i-s. 


!«»-- 


P"'^ 


canal  whose  hanks  slope  at  40*.  anrl  wliirh  has  to  duuiart  awav  75  1  uliic 
loct  with  a  iiUMM  vcloiily  of  3  (t.  per  m-<oiii1.- 

■.Iiis.    Doplli    -  3.0  fl..   uidlh  at  ijottoni  =  2.62  (t. 

23.  TiiO  >eclioti  of  a  1  anal  is  a  rc;;uiai  trapc/oid  ;  its  slope  is  I  in 
jtx);  its  widlli  at  the  hottoni  is  S  ft.;  the  ^i'ie>  .ire  inclined  at  30  to  the 
verti<al.  On  one  o<<asioii  wln-n  t lie  water  was  4  ft.  deep  a  wind  -vas 
blowiij,'  lip  tlie  canal,  causini;  an  air-rcsist.iuci'  for  each  unit  of  frci.  snr- 
f.ice  e.pial  to  line  lifth  of  thai  foi  like  units  at  the  bottom  and  sides. 
wIktc  the  co;'ll"li  icMl  of  fiicli.jii  may  be  taken  to  i)C  .08,  |)elerinine  the 
dischari-e.  ■'">•  75-34  <  u.  ft.  per  s.c. 

24.  .\  canal  is  20  fl.  wide  at  the  bottom,  its  miU'  slopes  are  1 J  to  1.  its 
loii-itudin.d -lopi-  IS  1  in  300;  calculate  11. M. I),  and  the  How  per  minute 
across  aiiv  K'''ei'  vertical  section  when  there  is  a  depth  of  S  f;.  ..f  w.itcr 
ill  the  (anal.     (Coelf.  <<i  friction  =  .uoS.  1 

If  a  weir  2  ft.  liiuli  were  built  across  the  canal  what  would  be  tiic 
incri-ase  in  the  di-plli  of  the  water.' 

.hi,.  5  .'4  ft.:  27f>2.777o  cii.  ft.  per  sec;  2.79  ft. 

2;.  Ill  the  ( )uiri|  c.m.il  the  c.irllien  b.iiiks  sh'pe  .It  <  <it  '  l}.  and  the 
bottom  wiilth  is  3.5.  Find  the  deptli  of  the  water  when  the  discliari^e  is 
3000  litres  per  second,  the  slope  of  the  canal  Ikmiii;  .I23<)  per  1000.  Also 
find  the  mean  velo<  ity.  ./«>.  i.j  ni.  to  1.4  m.;  .4  ni.  per  sec. 

26.  The  banks  nf  a  raiial  slope  at  45'.  the  >.<  lion  beiiifj  a  trapezium. 
The  discliarv;e  is  to  be  1  2o<j  littes  per  second  at  the  rale  <if  .5  m.  per 
SCCtJiid.      I'iiid  the  liest  b.  .tloni  VMdtli  and  depth  .iml  also  the  slope. 

,/«,i.  .>y|  III.;  1.14  III.;  .oo<J4  ai  cord  1  lit;  \-    li  i/ir  and  .0003  accord- 
ing; to  Manning,  the  mean  beiiiK  .0003;. 

27.  In  the  transverse  section  .//.'cV^of  an  open  channel  with  a  ver- 
licl  slope  of  1  in  300.  the  bottom  widlli   is  20  ft.,  the  an^le  .l/>'C    -  <A> 
and  the  annle  /lt/>  —  4j".      Find  llie  heinlit  to  which  tile  water  will 
rise  s<i  lh.it  the  velo<  ity  of  llow  may  be  a  maximum  .  also  find  the  d  s- 
l•|lar^;e  across  the  section./  beiiii;  .ooS. 

.»««.  1 1.71  S  ft.;   1584CU.  ft.  per  secoml. 
2.S.  Tlie  sewers  in  V.iiicoiiver  are  sijii.ire  in  section  and  aie   l.inl  with 
one  diagonal  vertical.      To  wh.il  lieit;lii  should  the  w.nter  rise  so  that  i./i 
thevchicitv  of  flow  iii.iy  be  a  m.iximum;  i'>  '''•■  ili~.  h  .-.•-   m  ,v  b.-  .1 
maximum.'     {  A  sidi- of  the  sipure       I2in.i 

.  f«v.  III)  .2i;2  ft.  aboM-  II  ■!  i/c.nl.ii  in .1 

(/''t.57.)7  f<-     ■■ 
;>>.  The  section  of  a  ch.iiiiiel  i"  a  rhombus  with  a  diagonal  vet tu,i!. 
llow  liii;h  iiiiist  liie  w.iter  fis-  in  the  cli.iiinel  (.fl  lo  j^ive  a  m.iximuin  ol 
flow,  and  ih  lojjivc  a  niaximuiii  di!«:li.ir^;e  .> 

.Int.   It  /'  is  the  lfni;tli  of  the   liori/onlal  diameter,  and  if  '1  is 

the  iiii  lin.iliiin  of  a  side  lo  llie  vt  rtical,  the  water  musl  ri*e  aliove 

llie  lion/. lilt. .1  di.iineur  to  the  liri^;tn  /■  .        '  -  ;    and 

I, .  the  height  />  cot  '<   x   .4t)<)'»  in  i/l. 

30.   .\n  ,i(|iiedilct  lias  ;i  niveii  A<>\>"  .old    .  .ijuare  htHioii  Willi  .i  lii.ijj- 


ySi' 


i:.\.-iMri.h.s. 


33 « 


1  ,..r,iril      Show  that  the  <iischa,ne  at  niaxunu.n  velocity,  the  .1,- 
:;;:;;;  .1^^  <>.»  -.■  t,>.  ..axinuMU  ..i«.har«e  arc  "Yhe  rat..s 
Trv     !niU,   ...4-  ana    that  the    .■..rcs,...n.hn,    n.oan  .icpth^  ar. 
I     ,-  ■     J  ii.-itv  a  side  ill  the  siiuarc. 

f  ifr  uT-f  iHM.ht  A.  is  la,.l  w,.h  us  base  horizontal,     toinpau- 

iS,:';:;"!'::;.^:::^: ..  >  w...,  the  ..ua,.titv  co,.v..ve,. .  a  „..,.., 

and  rm.l  ilic  ,urre>i.oii.lin-  mean  d.-plh-. 


../«.(.   fjiiiintilii^ 


■  \  ■■■ 


!.L)  lO'. 


(.1.2 1 8//. 


.,     V  length  uf  a  crcvdar  aqueduct  of  wateuvay  ./  a-,.,    n.eau  .U|.in 

.'     ,       .   r.    1  .  V.    hv  a  le,n-.th  ■<(  an  e.umaleut  rectangular  aqueduct. 

H  ;;:  d:;u::ru:.^:' air  .,::...  „.  w,.ith  ..  t...  rectan„dar  section  ... 

show  ih.Ll 

„„.  ,.„....  .,,  -'  he..,,  .■  ...r  the  ,.,,e  andM..  th-  rectangular  a.,uedu.  t. 

\-,/,       III  tiisi  aplirnxiliiatiu.is  take  /■        '  ,    .       i  . 

^      Til!«    O.c  Cemcient     .    f-  r    a    ,.ven    open    channel    to    he 

oooHX>5«  and  the  Co,  res,,on.i.u«  ce.hcie.U  (=  ^  )  f-r  ,iK-«  -   .■  he 

.«.,.  .„SS  show  that.  a,.pr...x..nateh .  A  the  v-.lu.ne  of  flow  u.uler  the  san.e 
head'.,  the  .sa.ue  huth  (or  the  chan-.el  ..nd  ...e  ,..,K-. 


,/'/• 


8.  '■'. 


,  ,„„„  ,,c  sectional  area  ..<  the  water«av  in  the  chan-cl.  J'  the  «e.ud 

'"■"'"•^;5^::\h:'::;e•::HL::er:::n  the  ,..eced.n.  exan„..e.show 

34.  U-.ni{  tilt  sanu  ,„.i,rlv  SS  per  cent 

.,..,  „.e  lo>so(  head  per  ..n.t  '    ^-^'  ^  ,P ';„^;     [^  ,.;,.,,..|,  ,,,,,. 

Hrealer  than  the  loss  .n  a.i  oiK-n  ^cnl..  ..,  ..i..r 

wav  and  u.v...H  the  sa.ne  d..schar^;.^  ^^ 

.   ^''  .V    I,H  «ct.on  i^e,.l;-sh..ped.  the  lowest  iK.ri.on  he,n«  a  sen>. 

"'  ','"  ';'•'-",,....      The  :q..cd.laconvo..  ..ornu.lly.  .-k.  litres  p- . 

""  :;;    n.,     the  ^  Se  .....enlh...  at  the  centre  of  the ,.rle  hv  the 

v.o.u      ''•"     '7      ;,.,,,,,,,,,,.., ,,„a,  area  of  the  waterv.ay.  .t*  deplll 

:,r;;:r:;.;:::tv  oMi.-  .<«..  .m-:  m  ^... .-  -^ «-  3^^-  •-  -. 


IP 


4  r 


35^ 


ill 


i  I 
1  t 


W^ 


I  sin 


'ttt. 


Dcii 


ti.w 


i:xAMPt.h:s. 


lorinuLi   for  a   (uciil.ir  ■iquiMlurt   of  radius 


,-|u-li  till-  wc  tti  (1  per. meter  subtends  an  .iiij^U-  ■  p|  ^40°  al  the  eeiitre. 


Alls. 


.2(<\H>\ 


37.   A   circular   acjacdiict    ul    '1.56  ft.  diatn.   eDiiveys   4().44  cu.  ft.  u{ 

goo.     F"ii,(l  iiJ)  tlie  aiij^le  ?ul)tcnded 


w.iter  (KT  sec. 


TlK 


lope  IS  I    III    10 


at  'he  centre  by  the  uater-liue;  {/>)  the  cle.ir  lu'.id  .il^ve  the  u.itei  ■sur- 
face; (r)  tlic  velocity  oi  flow. 

Alls.  (iM  240'  30'  ;  (  f>\  l.«)3  ft.;  (. )  I  Si  5  ft.  per  sec. 

38.  The  Avre  ( ireiiiar  aqueduct  conveys  j.05  cm.  per  secfind.  and  in 
one  lentil  the  slope  is  4  in  10.000.  lis  waler-liiie  subtends  1:0°  at  the 
centre.      Find  the  radius,  taking  b  =  .oooj  as  a  first  approximation. 

The  surface  has  a  very  sino;itli  co.it  of  cement  .02  in.  thu  k  ;  de- 
tirinuio  the  actual  waterway,  tlic  welled  perimeter,  tlie  mean  depth, 
ihr  velo(  ity  of  flow,  and  llie  clear  hcinlil  above  the  water-line. 

Alls.  Radius  =  .88  ni. ;   1S13  sq.  m.  ;  3.54'^  m.  .  .51   m. ;   1.13  ni. 
per  sec. .  .443  ill. 

39.  The  Potomac  aque'Ui<  t.  wliirli  is  f.  ed  with  lirick,  has  a  di.imeter 
of  9.0:25  ft.  ami  a  slope  "f  .143  m  10,000.  The  water-line  subtends  an 
angle  III  240*  at  the  centre.  'I'.ikmn  h~  .oooa<K)i^,  detei  mine  quantity  of 
water  conveyed  in  t.;alloiis  f)er  d.iy.  Aiis.  69.91^7.071  Imp.  K^'H'ms. 

84.ol9,o6C)  U.  S.  ■rallons. 

40.  Takins.;  /'  (Xxx)6o9,  lind  the  aiij;le  subtended  al  the  centre  by 
the  w.itei-linc  and  also  lind  the  free  height  above  the  waier-siirl.ice  in 
the  V.iniie  afiueihict  when  coiivevinn  4'*-44-  <"■  ft-  P""  sccoiirl,  the 
di.iineier  of  tin'  aqueduct  bein>{  6.562  It.,  anil  the  slope  I  111  lo.ocx). 

./«!,   240    50'. 

41.  ShiiW  that  the  quantities  of  water  conveyeil  by  a  ciicular  aque- 
duct of  radius  r.  when  the  water-line  subleiuls  an  aii}4le  of  240  .it  the 
centre,  when  the  velocity  of  flow  is  ^jre.itest,  when  running;  full,  and 
when  the  (piantily  conve\ed  is  a  iiiaxinuiin,  are  in  the  ratios  of  1  lo  1.0S6 
to  1. 131  to  l.lHS,  and  find  the  an^ies  siiiiteiuleil  at  the  centre  by  liie 
walir-Iines  in  the  tlirc  last  cases.  Also  deteiiiiine  the  mean  liydraulit 
deplhs,  \>n.  Aiix/>>.  257    27  ;  360  ;  30.S'. 

M.tiH  tiff<tlis,  .(*Jir\  .f)oS/ ;  .jr;  .573^. 

42.  For  .1  SMI, 111  t.K  iioiii.  ler  the  velocities  are  .16;,  .205,  .29S,  ,366, 
.61  metre;  the  number  of  levoliitions  per  second  are  .6,  .835.  I.4fi7, 
1,805,  V'4--     I'liid  the  constants  Lorri'por.t.ini:  lo  tiic  wheel. 

^/«.t,  .K'9;  .061. 

43.  ,\s«uniini:  (ll  th..t  a  river  flows  over  .1  lied  of  uniform  resist. nice 
to  source.  (21  that  to  mainiain  >talMluy  the  velocity  is  constant  from 
source  to  mouth;  (3)  that  the  river  wclioiis  at  all  points  are  similar; 
(41  tluit  the  discharge  increases  iinifornilv  in  consrquetice  of  the  supply 
from  afBueiils— determine  the  longitudinal  section  of  such  a  tver. 

Anu  .\  p.ir.ibola. 


l£: 


It  n 


E\.iMrii:s. 


44.  In  an  aqueduct  with  a  Siopc  ..1  1  in  10,000.  tlic  (U-pth  of  waicr 
orif^poiHimg  to  a  condition  of  uniform  stca.iy  motion  is  1.77  ft.  At 
a  certain  point  the  dcptli  is  incr.ascl  t..  4  4",  't.  In  a  wcir  377  ft.  in 
liei-ht.     rind    the    distance    to   which    tlic       rise'    e.xtends    al.,n-    the 

■^  ,   „,  A/IS.  50.038  ft. 

aqueduct.  ,        , 

45.  The  channel  of  a  river  32S  (t.  wide  is  narrowed  ny  the  abutments 
of  a  bri,i..e  to  a  wxitli  of  4:1.05  ''•  T'le  depth  ot  the  water  under  tlie 
bri<lj;e  is  i^.f.J  ft.,  and  the  quantity  of  flow  per  lioiir  is  :.4of..25.)  gallons 
Fin-'the  height  of  swell.  ■-'"^-    '"•*  '^• 

46.  In  .1  broad  channel  of  appro.timately  re,  ia.;nular  section   there  i- 


Alls.  .104  fl. 
UM^n*.  cji  ^\%elJ.  ^ 

,1  broad  channel  of  appro.timately  re,  iai;nular  section   there  is 
anall   chaiijie  of    //V    in   the   depili.      '^'' 


in   tiie   (lepui.      Show    iliat   the   correspoiidin;; 
.if  fiow  and  in  the  disrharije  are  J«?  and    ij«- 
respc"livelv.     Also,   if' the   banks   slope   at    ui   anRle  0.  siiow   that  tli. 
•    .  1  .       ,  ii/.iii  i/i  1 


changes  in  tlie  veioi  ily  of  fiow 


respectively.     Also,   if   the   banks   slope   at    ui   anRle  0.  show   that  the 
chan^-es   become    ^^^,^-  ^,^,„, J    •'""     ,00^2./        /■^m'l)  ' 

tively,  //,  *. ./.  /',!'.  iui'i  'J  ''^''"^  t'x'  '"''•'•''  <li^^l>t''.''i'^;'J'i'-area  of  water- 
way.'wetted  perimeter,  velocity  of  flow,  and  discliarye.  respectively. 


li 


ill' 


A  im 


i%  1 


("Il.\!'l  l-.K     1\'. 


!-;a\i^,  rKi:>- 


h 


liJ 


V     : 


Wi' 


\i  (TMri,  \  M  M<S,    \VATi:i<-PKI'SSCI<i: 

i.Ni.i\i;s. 


I.  Hydraulic  Rams.  l'i\  iiuan-  .if  tin-  ];}(lraulii  r.mi  a 
(]iiaiuit\  ol  \\atLT  l.illiiiL,  thiiniL;li  .i  \  citica!  ili'^tanCL-  //,  i>  iiKuk- 
!■!  Ii'nc  a  siiiallrr  wiML^ht  of  v^.ittT  t"  a  hiL;lu'r  lc\x-!. 

rile  water  i-;  lir"iiL;ht  trmii  a  reservoir  thrnii;.;!)  a  --ii])i)ly- 
])i])e  >.  At  the  eiii'i  iit'tlii-  pipe  tlure  i^  a  \'al\'e  opcniiiL;  iiUn 
ail  air-eliaiiil)er  (  .  uliiili  i~  (.  i  miuxted  w  itli  a  (lischar.L;c-pipc  />. 
At  /:  tlicre  i-  a  \\t.'i;^'litc(l  clicck-  Mr  claek-valve  uponiii:^ 
itiwariN,  ami  the  Ien;4tli  wf  it-  stem  i<<\-  tlie  -trukci  i>  ri'.L^iilated 
i)\-  means  (if  a  nut  <>r  enttar.  Winn  the  \\astc-val\e  at  /:  is 
•  ■pell  the  vvater  l)e;4ins  to  escape  with  a  \ehicit\-  due  t"  tlic 
head    //,   and   sudilcnly  closes   the   \.dvi  The   momi'iUniii   i>f 

the  water  in  tlie  pi|)i-  upens 
tlie  \al\e  at  /■',  and  a  ]iiir- 
tii>n    <<(  the    water   is  dis- 


/ 


^ 


1     ' 


P 


!X 


txiif^g 


Cll.iru;ei!  inti  i  the  air- vessel. 
Frum  tills  \csscl  it  prissfs 
into  the  iiischar^'e-i)i)ie  in 
consccnience  of  the  reac- 
tiiin  i>r  the  iumpri" 


-^vT^^* v  At  the  end  of  a  vcrj' siin 

!  inters  ;il   of  time  the    im 


Kii;.   !7p. 


rt 
)- 
mentiim  (>ftlu'  water  lins 
lii'en  destroyed,  the  \al\e 
ni)cninj;  into  the  tliaml>er  ('  closes,  the  waste-valve  a^Min 
o|)ens,  ;.nd  the  action  conm'.eiues  as  before.      It  is   fMimii  that 


HVnR.-U  lie   I'RIiSS.  .i.i."> 

thr  .ftuinuN  Mf  tlir  ;ain  1-  iiKi.M-c<l  h\  mrr.Mluun-  the  small 
.Lir-\<-rl  /.  1  lu-  \\,i\x-inntion  surl'--.!  up  m  the  -upiiU-iMlK- 
1,\  thr  oiHiim-  .111.1  dM.in.L;  "t  tlu-  v.ilvc  ..pcniiiL;  nU,.  thr 
ch.nnlnr  (  .  li.i-  Ih'imi  utili/ol  m  drivini;  a  pi-tnti  so  as  to  pimip 
u]i  wati-r  lioin  sumc  in(k'])ciuli--nt  soiirtc. 

l.tt  ■•  l»  tin-  wlocitx  i)f  flow  in  the  supply-jiiiR'  .it  the 
niMiiKTit  \>.  lirii  tin    \alvc  at  I.  is  closed. 

l.c't    W  'k-  the  \veii;ht  of  the  mass  of  \i.,iter  in  motion. 

■1  iKii        ''  '    is  the   enei-\   of  the   ni.iss.  .iiul   tliis   eiier-.>-  is 

,S  " 

rxjHn.Ua  in  openm-  the  valve  .it  /.'.  torciiv.;  the  water  into  the 
.lii-eh.imlHr,  onn.iresMii-  the  .m .  .md  linall>-  eausin_L,'  the 
..levition    of  ,,  uci-ht    \\\'A   the  u.iter   thvou-h  a  vertical   dis- 

t.mce  // 

l.,.t     ,,      1„-    tlu-    head    c->nsume<l    in    tVictioiial    .iiul    "th   r 

hydranh     resist, nues. 

Ih.n 

ir,  r- 

If,//    i.  //,|  :     tile    .ntu.il  uork  done  =      ^   -. 


Mil,   e.iu.itioii    sli,,us   th.it,    houe\er   -i^'at  //    !na\-  he.    il'^, 

h.i-    I  definite   .ind   jiositisi'   value,  .md  therefore  water   m,i\   he 

r.iised  to  an\    reqiiir<ii  hei-lit  1)\    the  li\ilraulic  r.im. 

W'Ji 
The  efficieiicx  of  the  ni.u  liine 


,,.',  ,  ,ind  ni.i\'  he  .is  much 
","1 


as  6('>  per  cent   it   the   machine   is   well  m.ide       Accordini,'   to 
d  .\ul>iiisson. 


"•/', 


\^ 


2.  Hydraulic  Pres.s.  llu  hvdv.uihc  press  is  .1  machine 
l>\  means  of  w  Im.  h  i.;reat  i)rcssiircs  can  be  e.xerted  and  liuavy 
ueij^hts  lifted,  the  encr^'v  bein^  triLnsmitted  tiuou-h  water. 
It  consists  essentially  of  .i  strong;  cast-iron  or  cast-ste  1  ch.i.n- 
ber  or  cvlmder  containing;  a  plunder  or  r,im  wliich  is  ;ieicd 


330 


HYDK.ll  IK.    PRhSS. 


b\ 


upon  l.v  natcr  p.mipcl  ihmn-h    pipin;.;    into  tin-  .h.unbcr  b\ 
snu:l.-,utnu:    tnac-p^;n,p,    which    nun     Ik-    c-.Uur    'a  nrkcd    by 

hand  or  Ir,    jiowrr. 

The  .ution  of  the  prcs-^ 

(lfpcn(K  on  thr  i)i-incii)!e 
that  lliiiiN  pn-.-;  (Miii.ili\-  in 
.til  clirfition-;  an<:  ihu-  thc 
prossiirc  \>cv  squan  null 
(in  the  ram  is  c(mal  to  the 
prc-^-^arc  jtcr  s<|uari-  i'uh  on 
the  p.inni)-pUin;4(  r.  <  'ri:_;i- 
iially  (liscovcrcil  b>-  I'ascal, 
the  inc-s  uas  first  made  of 
practical  utilitv-  b>-  l'.rani,.h.  -aIio  madr  ilu  niovin-  parts  u  atcr- 
ti'jlit  li\-  the  introduction  of  ^  up-lratlur  packni- . 

■  liar    '  ■!    n  I""'"'  ^'^  !"^'' 


I'M.     I-' 


111-  r.iin    I-'    1 


)ail<cd  w  itli  a    k-aturr   c 


r 


is  filled  into  a  ncess  turned  out  in  tlie  neek  .4  ihe  .\lnulcr  and 

is  kejit  in  place  1)>-  the  e\liniler-eo\  er  -land. 

Aeeordin.L;  to    ixperiinent-    niaile   b\-    iluk,       | 

the   liietion    at   tlie    eoll.u-   incrca-es   dn-eell_\-      ) 

with  the   diameter  of  tlu    ram   and  \\  ith    the  '^"-  '*'■ 

pressure,  hut  i-  in.iepen.K  nt  of  tlv  d.ei.th  oi  the  collar.       Iliek'-^ 

l.iu    .,t  tnetion  i-  eNjires-cd  1)>    the  f.nouin-   tormula; 


Ih 


r  lot  d  trietional  resistance        .o;i.p//'  or  .u4;i,//, 


aecordinu  as  the  leather  is  in  i-.od  eondition  ,md  well  lubri- 
cated or  is  new  and  b.idls-  lubricated. 

The  fnctinii  is  about  i  pM- >  ■•nt  of  the  im-.  ,sure  f  .r  a  4-in.  rani. 

At  1<!W  pressures  //<•;;//'  f'thkiuj^  is  inv.ui.dil;.  u-^-A.  .md 
sometimes  also  for  pressures  as  threat  as  zooo  lbs  p^i  s,|  „,., 
but,  -enerallv-  spcakin:^,  it  is  rar>l\-  used  for  pressures  ,  x^ed- 
\\v.'  about  -no  lbs.  piT  s(i.  in.  i  he  ram  is  .Iriven  lorw.uds  b\- 
thi'  pre-sure  ot  the  u.iter  throu-h  the  ti;;ht  eollar.  an-l  is  capable 
of  liftiu'.;  a    ueiidit  or  exerting;   a  pressure    uhiJi    is   limited  m 


U<i 


HYi>K.>i  lie  ri<iss. 
..„,.„, .V  .V  U,e   .....—   ^' -  '  — 

,^.^    ,|-b.   the    uc.^lu   nntlunnn,  Muludnv. 
tlic  v.un. 

-,„,.„   ,/,,    :.v./  ./'.".-        "■'        4'''" 

-  .  ,    ,  ,1,-,  ui.  .-   -'"""  tl"    '"i"'"" 

;;,:.;::::,;, ui.u.,.,...,..-,....^"'^  "- ■ 


or 


/•,  tlu'U 


./',-/'. 


,S.-c  Ap,..n.l,x,  •■  Th.  ..f  S.ruaur.> 


n„„v  ni   U--   ni.n'.ituiti.in 


"  UovfV.) 


,,•  tlx'   Hnimah.  .ur  rnipinv..!  t..,  .1 


,   ,,„KtciU  ilcsi-ns.    but    XV 


,1  hum-  oi)cr.iti<.ii- 


\-\,r  i  \  ir.iuh:.  thry 


variety  .-I   pn-Mir.  ■>,,.,....   .     ,  .  „,„„  ,,,,,k,  „, 

v'""  ;':;::,:;.:r ,.:;:;:,  ,.'.„„.  .,.„„.„.,  ..«!»„. 

tlu'  inoiK-in   -'>  ^It-Ill  1 


//>/)A'  //'/'.    '  4i:k. 


iK.inn''.    w 


cUliii",    .nil!    l)iMi.lni:^    .K-pcn.l    upon    tlif    ]K-ciiliar 


icUaiita-x-  i.f  liN'lr.iiili'-    !)'i\^<-i   t"i'   -"^'i  I'ln]). '-^  ■ 


I\(lr,nilic 


nrc-.si,-  (nr  r.-r-in:-;  lia\c-  lar-rlx  -iiiHiM-ik-il  the  -tcatn-liainincr, 


if  ') 


m-  . 


1^+^ 


mi 


i 


Ilvilraiilii    I'ri'-s, 


1  1...    1~V 
I'orlalili-  K'\  Ltf  r. 


ri.;.  i«4. 
Ualiiik'-l''"'"^^- 


and  it  i-  PMV,    >Mininon    t-  Miul    i.r.s..-^  uith    .ai;a,iti(-   laii-ini: 

tp  m    4,H,M   t .<)("'   ton-,  tlu-  werkin--   iiit<  hm!;.    ■  .1    i-rt-Miic 

l.iin-  as  L;ri-al  a-    ;  ton-,  prr  m].   m. 

Tlu-  Indraiilu-  jaJs.  1ml;,  iS;.  i^  a  portat)!.  nuu  liiiu;  for 
lai-iiiL;  lu-a\>  u  <M;.;ht-  tlirou-h  -lioit  .'i-taiu-t.--..  It  i-  a  coin- 
p.ut  c.iml.ination  of  a  for.  <--punip  ami  a  prc-s.  1  h.  ram  O 
tit-  thf  I'l-c--  .S  ,in.l  1-  i;'ad.  u  atcr-ti-lit  1)>  the  .up-li-.itluT  P. 
•11,,.  |„imp  is  worked  In  ttie  iip-and-dou  n  iiio^ ,  nunt  of  a 
liver  uliuh  pre--e-  upon  a  i  am  or  i-  eonneet  1  u  itli  otiier 
suitable  -eariir^  and  comniimieate-  motion  to  tile  j)unij)-plunL;er 
/^'  I'l,,'  water  ill  the  .  liaml)er  i-  tlm-  for^.-d  tlirniij,'li  a  \-alve 
into  th.' IndrauiK  e\-lnider.  de>  elopm-  a  pre-sure  wliich  ean^.'s 
the  ram  to  rise  and  to  lift  the  loail  restin-  on  tlie  hi  ad  //. 
•  In  A.iierii..  comi.rcvv,-.!  air  i>.  !ai,4e!y  "'^■-"1  f"r  1""^'  ''  =  "K.  fi\clini!.  ete. 


/ILLUMil.ATOR. 


y\-  tin-  inimp-pliin^'tT  ri-^i's  a  |).uti.il  v.Ktiiiin  i--  jn-nduci'.! 
ill  tlic  i)um]>-cli.iiiil)cr,  and  tin-  pn-ssurc  in 
the  rcsirvoir  />'  nVLrcomcs  tin-  rc--i'it;incc  nt 
tlic  ^I'riiiL;  'in  the  inKt-\.il\i'  ami  opens  a 
passaLjc  tor  tlic  watrr  into  tin-  puni]i-i  li.mi- 
htr.  I'o  lower  the  jack,  ,i  ri.-lic'f-\  al\  c  i-. 
unscrew  Oil,  ami  the  water  returns  to  tin- 
reservoir   />'  wliile   the   ram   falls.       Ilie   ram 


Fig.  185. 


Fn;.  1*6. 


ni,i\  'le  pre  iiteil  from  turnini^  rouni!  1>\-  ineiuis  nf  .1  steel  set— 
jiiti  -.(.rewed  on  the  ^jd.e  of  the  pre--  uu!  tittin-  a  \ertKal  slot 
in  the  r.ini. 

rile  t  i.ii-tnKtioii  and  action  of  -die  piiiu  liiii--l)ear.  ^■i^^. 
iSf'.  arc  essentialK  the  -anie  a-  in  the  hvdraiilic  jaik.  li\ 
actuating'  the  Icm  r  /.  tlu'  water  pa— es  into  the  hydrauhv 
cylimler  C'and  h\-  its  action  forces  tlie  puiui!  /'  down.  The 
punch  1-  rai-i  !  by  first  openiiv..;  a  relief-\  .iK  c'  and  then  loUL-r- 
ini;  tl;c  lever  .'/.  which  cause-  the  c.uii  to  raise  tin  li\ilraulic 
r.mi,  and  tlic  wativ  trom  thi'  h\draulic  cylimler  I'low-  l)ai  k 
into  Oic  reser\<>ir.  1  he  ichet-\  .d\e  is  now  closci!  ami  the 
]nnKhiiv_;  nperatioti  ni.iv   he  aL,Min  repeated. 

3.  Acctimulator.  low  pres-ures  of  170  lbs.  (—  392  ft.) 
to  250  lbs.  (  —  57''  tt  I  pi  r  s([  in  *  ,111  sometimes  b(  obtained 
from   a    natural    siip[)l\    or   from    a    icscrv(jir,    but   tiie    In^diet 


m 


ti 


^l 


h:     P*"^. 
.*»i-" 


i 


34° 


.H.Cl  Ml  I  ATOK. 


pressiir.-^  of  ;'Xi  ll>>.  (--;  K'mj  It,  t..  moo  Ih-.  =  2.V'4  ft) 
per  sq,  ill-  .uul  ui)w,ir-U.  ■aIiuIi  a\c  .iiiiiM,t  c\c'lu-.ivcl>  .idiiilcd 
to  the  uorkiii-  111  intcrmitti'iu  iii,ic  l-.mr-,  v.\  \^\  he  .iililKi.iIly 
;)n)duccd  b\-  incati-  ot"  puinpni;_;-Lii-iii<-  In  .i  dirra  -uppl_\- 
t!ic  capacit)'  of  thc^c  i  n-int---  must  In-  >uirKnnt  t^  nui  t  tlir 
maximum  dcinaud  at  an>-  lUMmiMU,  l)'it  tin  <1iu  tuition  in  the 
dcniaiiil  upon  tin  main-.  I'm-  ^r.uic^.  cap-tan,-.  ili\  at<  >i ,,  c  ti  , 
was  -'iiiii  foun;i  to  In-  -()  L^i-rat  a-  to  niidc!-  mipcrativc  -mhi,- 
metli'ui  ..f  vA'/v//;'-  i'ncr;_;\'-  llus  ha-  !ici.!i  I'llciti-d  hy  tiic 
iutroihicti'Mi  nf  the  at\u)iiiiLitor.  whuh,  ni  it-  -implcst  r.irm, 
roiisi-t-  of  an  amiular  fvhndi  r  I'i--.  i  S;  partial!)-  or  w  holl>- 
filled  with  -crap.  -'.i-;.  oi  other  Iu-a\  >•  miti-nal.  or  ..f  a  -tries 
(if  travs  (I'll;,  l''^.'^  loaded  with  pi-  iron  or  lead,  -npported  hy 
a  cross-'aead.  on  the   t-'p  of  a  laiu  woikm^  in  a  eyliiuler  with  a 


tr^^ 


^1 


i  1  _ 


Err 

I 

/r' 


r: 


:rij 


\ 


Fui.  H7. 


Fl.i.  i<?S. 


stuftiiv^'-liox  and  -land  at  tin  upper  end  '1  he  pre  — iire-w. iter 
is  adiuitteii  li\  a  liranJ.i  jupe  at  the  lower  end  .\\v.\  rai-e-  the 
ram  to-etiier  witli  iheuei-ht  it  e.urie-  lliii-.  if  H  '  ton-  are 
lifteil  tl'.iou-h  a  vertical  di-tiiue  .^  and  if  tiie  u  ater-pre-surc  oil 


lt<T-:SSlllf-:K. 


M» 


tin;  ram  -f  </in.  -UanuU-r   i^  ^  Ih-^.  ]>vv  •^■\.  in.,  tb..    T^a,.;  -t.^rc 
(if  c-iici  ,'\  111  ti.dt-poumU 

=   2  240  1 1  -V  >/". 

Wlun  -hr  a..mnuiat..r  ha^  rc.ulu'.l  tho  hi-hc<t  Ji-nnt  it 
actuaU-^  a  Uacv  v.liuli  ■^hut-,  -U  tliL-  -leain  m,  tliat  tlic  cnu"'^- 
,e.a^.  t..  .v..rk  ana  tlir  ac. mnulatnr  UU.  Wlun  it  lia^  na.'u-,l 
thr  Inu.-.t  !>-nU  it  a-ain  aetuatf-.  a  l.-vor  vvhi.h  ..p.n^  a  valx- 
;,n,l  a.lnr.t^  ^trani.       \  hv  eni;inr-  a-ain  o.inuitMU  <■  tn  v...rk  an.! 

tin-  aCLuniulati  >r  li^c-;. 

In  -niaa  I'lant^  tiu-  a.  onnulatM-,-  lulU-  provuk-^  Un  tiu' 
i/, ',,,->  -t  -vitTuunt  Mur-y  to  meet  tiv-  nKuncntai}  tliu  tuaiu  .n. 
,,l    ,lcniana    fnr    tlic    p-urr    nrc  r-ar\-  t.  ■    u  nrl.    niaduiu-v  whuh 


;„c  , litem. .tt.-nt  m  a.u.m,  and  u.th^.at   thr  a.,  oanulatur  panip- 
in--.n-iiK-  -f  -rcitti-    ..ipavit)    u-iilcl    Ik    rciuin   !.       In    lar^'c 
piant.ra.  .n   the  >  Uir-   ..I    l.on.i.,n.   ManJiL-tr-,  an-i  <  .las^-.w, 
the  total  ae.unuilatnr  ^.tma;^.-   .  apauty  i>  a  vciy  mimU  fraLti,.,, 
,,t    the    total    supplv.    and    at    the    time,    uhen     the     .lemaiui    1. 
heavv  the  ae.unn.iator,  are  usuallv  ahno^t  ^t.ti.  .nary .       in  Mieh 
cases  the^    mav  he  .onsi.lere.l  rather  as  nx„/.a,.  .s  ot    pr.  -M,re. 
'Ihey  are  aWn  of.jreat  importance  in    auto„,.,tu  all}'    ta.  1  ..titin- 
the  eontml  of  the  plant,  ami  a.t  aOaiUer.  m  pr.Nentu,,,  l.ieal.- 
a,H'  an.l  ^hoek^.       ll  laek  of  .pa.  e  prevent^  the  n^e  o!   an  .aa  i - 
imilator    ol    the    tv  pe    in^t    .le^erilud,    an    /«/,.. {A- r.    I' :^     '  ^- '• 
„,av  he  emplovc.l.       Water    at  a    pre-Mire    o,    ,Mhs.   per  m;     ,n. 
is   admitt,   1  from  the  uater-mam^  or  from  a  tank  at  a    MWtalne 
elevation  to  thelouer-ideof  a  piston  < 'I  diamet.r  /'ms..  ..ork- 
i,,.,n    anhvdraulu    olindcr.      The    piston-ro.l   .,f  diamet.r    ,/ 
ins     forms  tin-  ram  of  the  ac.nmulator   /;,  and  NVorks  thron.h  a 
N^.aer-ti^htne.k.       ilms  the  pressure  m  the  aee  um.ilator  m  11,.. 
iK-r  s(i    in. 

"     n,i'  d^' 

4 


11 


N 


'iJv 


w\ 


1 

III 

1               .; 

1  : 

,\        ': 

|ii 

-P 


/>//     '  KLMI.-U      ".'  /   M!  /.■//')"'. 


and  tlii-  i-  also   the   iiitcii-ity  of  tin    i.ii>>urc   in    tin-   h\ilraulic 

main-^  (  . 

iurdiicli  -  ,in'i\yfiittiii  an  umu'i.it'ii ,  V'v^-  I'jO,  is  also 
dcsi^Mird  I'll-  lasc-i  in  wliicli  -pair  i-  "t"  nnp.  .rtnu  <•.  .\  lioavy 
c\liiiilci-  .1.  \vilb  the  usii.il    -land-  .md.  ,  up-katlu  r-.  at  the  top 


fids     *=> 


==>J 


Lri 


Fii..  I  ^It- 


Hi 


i^3si 


ami  bottom.  i>  loadid  uitli  a  luiintxT  .il  load  <>r  cast-iron 
wci^dit-  \\\  I'tttd  int(t  c;!-!!  .itiirr,  aii<l  didc-  np.iii  a  ram  /.'. 
fixi-tl  .It  lltf  iii)iHr  i-nd  1)\-  ,1  bracket  .iiul  at  tlic  lower  b\-  a  step 
A  brass  liner  is  slirunl-  ujion  tlu'  lower  portion  o|  tin- ram  *  -o 
that  Its  diameter  is  sli;,'lltl\-  1,'reater  tha'i  that  ■  .f  the  ui)|"i 
jKirtioii.  A  hollow  passa;;c  C  is  tlrilled  a\iall>  alon-  the  ram 
;iiul  connects  with  a  ero-;s-passa;;e  iii'>t  alxne  the  brass  liner. 
The  water  is  pumped  tlnoii^'h  tlu  inlet-pipe  /.  tdN  the-.- 
passaeos  ai'd  exerts  an  upw  ii  I  i)re.snre  n\er  an  effective  area 
equal  to  the  /////./<//<<  between  the  areas  of  the  lower  and 
iip|)er  iMirtion-  <>t  the  ram.  ihiis  \er\  Inav>  pressures,  up 
tn  2000  lbs.  per  stp  in  ,  or  inure,  can  be  rea»iil\' obtained  with 
a  comiiai.itni  1\    -mall    w.i-ht        lint    tiie  \(.Uiine   of   water   is 

*  Ttu-  ram,  Imwrvr;,  is  uiil.illy  tulid  >Iccl. 


.null,  ami  am  lap..  .U'.uan,!  t,  „  j.  .u  rr  u  ,1!  .  an-.-  thr  lua,!.,! 
oHn.Ur  tntal!  raimlly.  ->  that  aIuh  it  i^  lu-,u,.;ht  t^  ,,  .t  a 
cnn^ulcrat.I,-  nuT.a^.-  of  ,„v.,.n  ,^  ,l,v.!.,,uM  uhuh  ,.  n, 
a.lvanta-r  ,n  ,.nulun:^.  nvrtn,.,  >  U  .  Ih.^  npiK-nn...!  u^u'it 
is  o,n.uar,l  l>x  nuan^  ..fa  Jiain  -Aith  a  ivHol-vahc  ubuh 
cnahk-tlu-  H,H,tin,,,H.Mt,.,n^...  th.  .xlnul.T  L.  Ik-  aut.jmat,- 
call>    rc;4ulat.-.l 

l.ct    II'Ih'  ih.    /  .'.'/  •I'-'a't  \\'  i'^i.t   lilU-ii. 

1  ,t   /    h.'  tlu-  liMi..n  ..I  >  a  Ji  ..IiIk'  ^  iip-irathcr-;. 

I.c't  7,  .  '/.  \n-  til-,    .li.iiiu  tri-  ..f  tlu-  1..U.  1   .uii!  iiinxi-  P"rti..n. 

<i!'  till-  r.im. 

With  tlu-  -.x  in.U-r  ,it  th.  lurjit  i  .il-x.'  it^l.-ur^t  i...Mti..n. 
].,,  ;>  1„.  til.  intniMtv  ..I  i.n-M.nin  tlu-  ini.t -,.i,h-  /  u  hen  thr 
cyhn>'k-ri-n>in-,  .uul  ^ ,  tlu- uitm-ity  u  lu  n  H  i^  tailnur.       llfi> 

l\-ir2i 


,,..„,,     ,„     .ppp.xmut.-    u„-..-u,o     ..t-    llu-    variatir.u    .  .f  thr 
inti-ii  -It'-  <>t'  pn-^-m"-  i> 


i6F 


r     i 


a,l.l  thr  vaim-  ..t    this   ^.>l,.al....    i^    ....imaiily  In  ai    ..l-ut     1    i-cr 
cnUnl-ilu-pn-i"-^'    '■"    •'   "•  '"     '•'"""   •''"""  ■'  1"'   ^'"'   '"'   •' 

Kx,)cnnKMU  lu^  -h--^^tl  llu  rlTui.-iuy  nf  an  .uannmlat.,!  L. 
In-  .IS  hi-h  as  .,.s  ,,u  .cut.  1  l.fi  ^cnt  bciiiK  I'-t  ■"  >  l^."  ,;;nu; 
.,,,,1  ,  yn  ..nt  m  disdiarumL,'.  It-  t..tal  stnn-  ..t  .  ncr^y  is 
C.,nMMi..t.v.-!v  Mn.ill  am!  it  >.nnu.t  maintain  ..  -npiav  Lu  anv 
Irn.^'th  ..r  time,  but  it  pcwsc-L-  th.'  ^-^'''t  a.Kanta^;.  -.1  lum- 
able  to  u-c  its  encr-y  .it  ..  I.i:^b  rale  for  a  sh..rt  iK.ii.)d. 


H 


3-U 


HYDK.il  III.    IM.ISliS. 


KjjnTTi'r^^S^   •'      form  ( .1' lu  i  inmil.iu  ir  kni.un  ,is  Him\mi'. 


I  m 


II   -5^ 


J 

It!  .  ^  ^ 


i.,M.-ii-.F-"rP"'"lB  htcam  Aw.nmiil.itor.       .\  r.mi  /\    w'tks 

ill     til :■     Indr.uilii.      i  li.iinhrr     //,      into 

uliH  li     w.itii     !■-    tnm  i!    1)\-    .1    ;i.iir    of 

m^mi-,       A    i>i-tii;i    /'    1-;     .!tt,u  luii     to 

till    ii]i])(  1'  I. ml  "I    tlu'  i.ini  .mil  \^'>il,-.  in 

a   olm^lrr    -ii|i]iiU(l    with    sir, mi    ilini  t 

liMin  the  1)1.1111  -,      A'^  --oi.ii  as  ]iits~iirc- 

watn     i-~     --iijiiiluil    to     !i\ili,uilK      lu.i- 

I'hnirrx  tin-  laiii  ami  pi-^ton  |,.!1.  (.pi  iiiiv^ 

tin-    -train-jioi  t,    -o   that     -tiaiii     l)ass(,-, 

int..  the  cni^iiic-i.xhiiili'i--.        I 'u'  liiiini)-> 

'  "*       tluii    lonitiK'iKc    to   work     iiiil    loiic    in 

Ki<;.  191.  more    water    to    rc|>KH '■    that     wliiih    1-. 

heiii-  liiawii  oiV       This  aee'iimilatoi  i-^  ^ipciiall)  lor  u-e  011  siiips. 

4.    Water-pressure    Engines.    -  In    tlu-sc    i  ti-itu-^    water 

uiiilei  in-e-'iin.  1  -  ailmitteil  into 
a  -troii^^  cliamlier  or  (\lm:l<r, 
anil  ait-  iipon  a  )MMon  or 
p!uf.L;er    ill    preei-^elx-   tin     ^amc    ^        <£jp 


iii.uincr   as   in    the    ease   of    the  ^T^  ^■ 

-.teain-civ^Mne.       I  he  1  \  inulei   1- 

Ma«lo  of  ^nn-inei.il  or   of    last 

iron,  ami  its  thickiR-.s  /.  wliuh  j-j;^^ 

is    relatively    iarijc    on    anouiit 

of  the  wear,  ma\'   l)i     i  aK  nlati  il 

from  the  lonimla 

f  ins.        .no2j^f,ii-^  i  .:!5  i"-  • 

/,    lu'inn   the   ]ir. --lire  in  atmo~phev<-.  .mil  ./  the   ili.mii  ti  r   in 
inchfs. 

Tin- friction. li  lesistame^  and  the  jios-^ihility  <>!  -rvere  <hoi  k- 
are  incnaseM  hy  rai)i<l  motir..j  an-l  reversals  n\  motion  1 1.  n, , 
the  velocity  of  flow  in  tlie  supply-pipe  s|u)ul<l  not  exceed   1.)  (t 


Ki-..  192. 


HWh'  li  lie    lll'^S. 


vl5 


per    <rcnn.l.    .mil    pr.'i'.T-.l.h-    -.'iinul,!    Ik-    limitcii    t.>    «'>    ft.    jx-T 
sriniiil    Alt     M,  ]).   i;'>  .  uhilr  tin-  iilini'^cv -ii.'iiKl  h.o  I    ,i  lont; 


stroki 


11    !!!-.h:Ul< 


the    -troki-   i--   iisuail)-  t'n 'm    jA  to '>  times 


Kli;.  193. — Sfcliiui.i!  F.levaiion. 


1     <c^ 


Fill.  i<)4. — C'ri>ss-sfctii>n. 


F,,;.    ,,).:  _Krrii;hl  h  .i^t  I'm.   r/i-K.il.inciMl-rani  1..I1. 

the  <li.im.  t<i  "f  till-  (.ylindcr.  ;iiul  t!u'  mean  velocity  <>f  the 
I'lim^'er  i>  .ilxHil  I  ft.  per  second,  rarel.v  exceeding,'  So  It.  per 
niiniitf.  \h  the  water  is  practically  incompressible,  its  free 
.md  imii'  ili.it<-   passa^^e   -hould    be   iii-^ured    li\-  means  of  iarjjc 


4 


,546 


HYnK-n  lie  l.lhTS. 


IP  'is- 


^TA 


km 


llu'  h\  (Ir.iiilii  -tnin  lift.   1 


aii'l  u  i(lc--(.inii  ]iMit>.  An  iiiii»M-tatit  ;ulvanta<;o  connected 
will)  thi^  jjiopnix  iA  iiu(i;iii)rc-.s!bilit_\-  i>  Uuit  tlie  hydraulic 
n>i-tnuis  may  ln'  ink  linitri)-  nuri.,i--c.l  !)>•  sinp''.-  closini;  .1 
\al\r.  Thus  UM  hiakcs  arc  rt-ciuntd.  Init  tlu-  wau  1  contains 
within  il-rll'itN  (iwn  hrakc.  am!  an  ah^ohiii.-  control  is  pr'  \iilcii 
uhuh  ■^ccun-  the  lui^hf-t  ilr-ico  o|  satcty. 

■j  hi-  uattr-pii-^-uri'  iHLjnu-  is  necessarily  a  slow-niovin^' 
niaihinc.  and  1-  l)oth  i  uinl)ro\is  and  i(>-.tl>-  unless  actuated  by 
prisMiri-  ot  -icat  intcnsit)-.  1  hesc  en.Ljines  are  advnnt.i^a-ously 
•jniplovcil  in  u.irkin;^'  cranes,  hoists,  elevators,  capst.ins,  ilock- 
■^ates,  presses,  and  otlier  machinery  in  whici  'he  actio;  i^  of 
an  inti  1  niitteiit  character. 

I-.    i()7.  more   completely  utilizes 
th.in   any  other   the    ])roperties   of 
iiK.)mpressil)iht\    md   (hrect    pres- 
sure,   and.    ovvini;    tr>    its    <,'reater 
safety,   its  adoption   is   -.onietimes 
recommended  for  elewitois  of  con- 
sider.ilile    heii^ht.       I  nder    a    full 
io.id  its  efficienc)-  n»ay  be  as  j;reat 
as  93   i)er  cent       Tlic  --ix-etl   of  a 
suspended    lift   i-~   raiel\    le.s   than 
100   ft.    pi'i    minnt.-  atid    otten   ex- 
ceeds   500  or  600  ft.  jier   minute. 
between  such  limits  a  lar;,.;e  varia- 
tion   in    the    efticii'tuy    mi^lit    be 
expicted.    and   althout;!)   the   cfti- 
cioncy    under    a    full    lo.id.    evtii 
when  the  r.im-stroke  i-.  ;nulti|)lied 
S  iir    10  times.  ina\-  be   75    or  So 
per  lent.    it  m.i>    .1I-"  tall    Ix  low 
40  )K.'r  cent  when  the  load  is  li^lit. 
Ihe  chief  loss  of  ofticiency  is 
due    to    the    fact    that    the    same 
quantit)    of  pressure-water,   .mil  therefore  of  enertjy.   is  used 


iiyiiR.iLiK.  rsi.i'^f-:. 


.>4  7 


whctlicr  tli>  lo.ul  i^  lK,a>  -i-  li-lit.  V..;-i..u^  ,icvio-  h.uc  in-,  n 
:ul<.ptc<l  to  rcnic.lx  tli'>  .-vil ;  tlic  Icii-tli  -f  -trokc  in.iv  In- 
autnmatic;ill>  ].r-i.-rfi..iu-,i.  ;w  in  tlir  ll.i.ti.-  ..■n-itu-,  f.  thr 
wnrk  U.  be  iloiH-:  tlu-  !)rcs-,uri--w.it.T  lu.r.  In  .ulnuttr.l  (..,  ,i 
Piirt  of  the  stroke  oiil\  .  the  reni.iin  Ur  'oeiitL;  piovi.ted  In  the 
<Usth:.rL;C-water;  cr.ine^  an.i  elevator^  aiv  often  provi.l.-l  vuth 
a  lar<,'e  cvlinder  for  heavy  h.ail-^  an.!  ,i  Miiall  .>  Inid.  r  for  h-ht 
loads,  ami  for  the  -ame  jnirpose  a  Mn-le  e\  hn.K  r  \v;th  a  ihller- 
eiUial  iiistoii  is  s,  niutiiiie'^  iiseil. 

Other  iniiiortant  h)sses  of  etf u  lem  >■  arc  (hie  to  (,,,  pipe 
friction;  ■/>>  elbows,  curves,  etc.,  aixl  abrupt  chau-e-~  ..t  sec- 
tion. ,.  ,  the  friction  of  mechauism. 

Let  /•„,  be  the  mean  intensity  ot  the  pre-Mire  in  'he 
cylinder. 

I.(  t   »  be  tin    -troki-. 

Let  7  ,„  be  the  mean  \elocit\-  of  tlu    jiluni^er. 

'Ihen 

the  work  done  per  stroke       -      /"„,>; 

4 

the  qnantitv  of  nioti\e  uater  u^ed  i)er  stroke 


7t,/ 
4 


1      .7,/- 

""      ^     4    •-• 

I-  diiuble-  or  sin^de-actiuL;  t\-p(  . 

"    be   the 


,1 


according,'  as  the  en^jine  is  ..t  i 

Aiialvsis.-    In  a  <ini  ct-actini;  pres-'.Tt-en-m.    Ut 
sectional   area  of  the  \\.iilsuv.;  >  \  hn.lei     h'i-. 
|c>8 

Let  </  be  the  sectional  an  a  of  the  ^uppi>  - 

pipe. 

Let  J   -^  11,1. 

Let  ILbe  tnc  wei-^ht  .if  the  uater.  pi-toii,  ,ind  other  m  ip- 
rocatin^^  l).irts  in  tl>e  working,'  cylinder 

Lft  /be  the  len^'th  of  the  snpply-pilH-- 

Let  f  be  the  accclei..tion  of  tin  piston  Then  ///  i-  the 
actelir.ition  of  the  w.iter  in  the  Mipi)i\ -pipe 


I   :..     \r 


■\ 


\H% 


i     * 


Pit 


t.       ijsl 

!  i 


a 


ti  i' ' 


1 

1 

i 

i'  -' 

'H 

.   'i- 

i 

r 

■  '■ 

m 


348  HYiiR.4iii(:  /.V(,/\7r. 

'I'hc  tcii\c  rciiuiri.- !  tn  aci.  dciMte  tlu-  pisinn 

If 

<r  • 

and  tlic  t  ( 111  1  -jM  .iiclniL,  inc-^-urc  in  fci-t  1  if  v.atrr 

■J  he    t'lirtf    iLiiuiiL'ii    ti.    .UAc-lciatc    llic  u.ilcr  m   the  -upiily- 
pipc 

,r 
.•> 

and  t1ic  (.■orrc^pnndiii;;  pi-v'^^iirf  in  fcft  nf  water 

A 
Siniil.iilw  it'  /    i-   till-    '.ihl;!!'.   '4'  tin-  d.--^  liaiL^i-iiipc   .nui 


/I 


it'^  sfctii  mal  ana,  \hr  jiri-^^uri  -lu-ac!  dui-  to  tlu  inertia  "I  tlu: 
tli-;cliai"i;c-\\alrr 

IJiiua-  tlic  tiital    ])ii'>---un'    in  Ur\  .it'walci    i"i|iiircd  In  <>\cr- 
cnnir  liurtia  in  the  -uppi\ -iiijie  and  i  \  hiider 

Tlu;    i|nantit\-  -^  ■  '//   lia--    lieen  desii^nati d   tile    lenL;tll  ot 

Uoikiiit;  i\linder  e(iiii\aleiU  tn  tile  im  rtia  •  4  tiie  in(i\in^'  jiart-;- 
Lct  tlie  eii'pne  dii\e  a  eianls  "|  lailiii-  1  ,  an  I  a-^-^iiMle  tliat  tl'a- 
velocity  /  '  ^l  tlie  eiaiilv-pm  1-.  ap]i' "\imately  i  instant  Tlien 
tlic  ac'-eieian..  'I  tile  pi  inL,'n"  lien  it  i~  at  a  di-taiae  .r  froir 
i«;s  central  i"  '-itii  hi 


t   .1 


H)  PK.-tC  1 1<:    /;.Vu/.V/;'. 
ami  tlu-  pn.--^>urc  iliic  Ui  uu  rli.i 

v  w 


.H9 


gr'  ,wA 


n!  x. 


tlu'    wnrkili' 


J,,,     -      i„.     tllr     \eiiait\     .'i     ill'      pl!i'l-rr    m 
vsliii'lcr 

lA-t  //  i)i-  Ihe  \rlout>   ")  the  u.itcr  1:1  liir  ^uppls -i>i|h- 

Let  //  !)<•  tlic  \'  'ticil  (li-^t.uur  hctucni  tlu-  .u  i  nii:ul,itMr-rain 
ami  till-  iii'itiir 

Ia-I  /<„  1)1-  tlu-  unit  i;r<--,-.un-  ,it  tin-  .k\  umui.it-r-r.im . 

I.ct  /"  Ijc  the  unit  presume  in  tlu-  u..rkin:4  .\lnuk-r. 

'I'lu-n 
i,         ,ii  />         .  ■         \  lo-^-^e-    >luc     tM     IrictKin.     ■^mi.k-n 

■riu;-> 

A.  -  ^        ^'^  -  "^  _  yj  +  lo,,c.. 
:.■  2.C 

Tlu-  tcnr,  '     ~"     •    |(K-.(   .  im;u  1)<- •ipp''"^''"''''''^  '"^l"'""^'"' 

,11    tlu-    t..iiii   A'  '    ,   A'  l)c;i'-    tlu-  i'k-iVku  :i!   ni    hyliaulu'  re-ist- 
'1' 

ancc.      lliiKc 


p   -     p       ,,v'        KV 

2g        -!Kr 


(I 


w 


tlu-   trrtn    //    l.rin-    .li-.ri-:.;.u  .k  ^! .  a--   it    1^     iMi.iib    \ci>-    -in. ill    as 

A, 
cmiipaic-d   >'.itli 

T  hu-  tlu-  iMtal  pi.-->-arr-hra<l  in    f">-rt   minircd  to  overcome 
inertia  ami  tlu-  livdiauli'    r(--i-iaiu.  e- 


,     .    4   nl  X  •        r       X-)  1 .  .     .     .     (2  1 
gr' '    wA  J  i 


V 


35° 


HYnR.iri.K     F\i'jIM'. 


aiul  is  1  c])rc-^c;itc(l  b\  the  nnlmatc-  l)v't\vccn  tlu:  ])ar,ibi  il.i  ,.,/ 
anil  tlif  line  <i/>  in  l-'i^'.  190,  iii  wliiih  .?/,v."  i>  -i  vci  taiiLjIc,  ab 
representing'  the  stroke  2i , 


/■-'/   //■ 


till-  prissiirc  due  t')  iiurtia  at  tin    ciici  i.ttlic  -trukc.  and 


iii  ', 


the  pres-au't'  recjiiircd  to    (ixcrtoinc  \\v    lixchaulic  resistances  .at 
the  centre  n|  the  stmke. 


The   ordinate   het\\(  i-n   tlie   ]iaraholn  f)i!i^  and   the   Hne   /";'• 
reprisents  the  hack  ])rtssere,  wliieh  is  neeessarils'  Dropnrtional 

/" 

.e.,     to        ._,(r-  —  .,-'). 


to    tlic-    s(i\iare    ot    tlie     piston-^elocit\' 


llenee  the  effeetive  pressure-head  'Ml  the  iiistmi,  transmitted 
t.)  the  iiank-pin,  is  r ^]ireseniid  li\  th>-  1  irdu.ate  hetneni  tin- 
I  ur\'i's  ////;'  and  (<;/  The  diaL;raui  -hovs  tn.it  the  ;)ri---su!e  ,t 
the  end  111  tlu'  stroke  is  \(r\'  kii";.;c  .'.nil  ina\'  heeonu'  e\ei  ssive. 
It  is  theiefore  u-ird  t'l  intrnluii  n  lirt-\  al\  rs  1  M  air-  'essels  to 
prevent  \aoler.l  shueks.  In  iirtain  la^e-,  lii  >\\  v  \.  cr,  as,  r.;;  , 
in    a     rivetiuLj-inai  hiiu  .     a    lua\\     pies^uic    at    tin     end    nl'    tlu 


stroke.   )u-t    ulu-rr    it    i--   mo-,!    iut.K'iI    t..   I'lMM-tlic   nvrt.  i>   <  .t 
^rcixl   uilv.mt.i-L-,  .111. 1   tlu-i.tnu-   the    iiKitM    cHci  t  i-   iiure.i^r.i 
!)\    tin-  u--(-'  "f.l  ^ui.])l>-pil)c  ..l'  siii.iii  liiaiiictci    ,iiui  .lU  a.>  iiniu- 
l.it'.r  with  a  small  'A.itcr  >cctiMii  iIm-,    197). 
By  (.(lu.ilHJii  M    , 

-=:';•/•■        ^' ''^' 

Thi^  <pLT.l   :■  ran  he  re-iilate,!    ,u  will  1>\    tli<-    tiirnm-     >f  a 

(,  ,^K.  ,1-  111  tin--  manner  ihe  ludiaulie  ve-i^taiue-  ma>-  he 
iiulelliutt'v    iiu  rea--iil. 

Let    the    eii-ine    he    WMrk;-\-    -tcaililx'  under    a    pressure   /'. 

ail.l   ht   .,    he  th'-  -pe.'d  .il  -teadv    iiL.tiMii.  ■Ilieil 

and 

ii-efiil  resHtame  ..MTt.inie  h\    till-  pistiin 
(  ^;-  IrKtKiii  het\\eeii  pi-tMii  ami  a^  eumiilati  .r-e>-liniler. 
It  P  i-  (limim-heil,   the    spe.  .1  V„  will  he    -ii-lltl)    ilKTea-ed, 

2gp  . 

but   II'    11"  I  as(    I  an   it   exei'e'i    \   /     ^j^  • 

5.    Losses  of  Energy.— Hie  Inssc-    may  1r    mum.  rated   as 

f..i!uw  .: 

(,;,    //.',    /.,'ss  I.^.inr  to  l'i.sl,;i-fri.  tivH.    -It  mav  he  assumed 
that    pi-tMn-tVieti.iii   eoiisume-   Iniiii    10  to    20   per   e-'iit   ot   the 

total  a\  ailahle  \s  or! 

{b\    I  In    Loss  /. ,  did   to  r:f^o-i',\tion—'\'hQ    loss  of  head  in 
llu    ■  up])ly-pipe  ot  diameter  </, 

_  4.//i«7'V- 

(^l       -,s 

•llu    lo---   of  heail  in  tli<    di-JiarL^e-pipe  of  diameter  ,/^ 

4//  (//  .-'jj 


ill 


itt 


I,"' 


35.         lOSSF.S   OI--  FSTRGY   IS  HYORAUAC  FNulShS,  IIC. 

1  !,„,,:  llu-b.t.il  Ins.  uihc.ul  m  pMK'-lriclion  is 

III'/      iii'r/'\i-'  _  .■  '■' 

Th,.  l„.s  in  tlu-  relatively  ^hnrt  unrk.ni;  eyluulo  1.  very 
MiiiiU  an.l  may  be  .liM-e;_^ar>le.l.  ,,.,:„ 

,,  ,  //.  /..S.S  /,..  -/,-•  tr  lu.rn,,.  -The  Nvork  expended  m 
,n,,Mn-.;  Hu-  uat.      ui  tlie  supple -I'M'^' 

X"'    -' 
^^,„,  i„  ,„„v,n^  the  xs  .ter  in  the  .li.ehar-e-pipe 

~   A'"         -  " 
The  total  work  thu-  expended 


^^'■•■'S.  ^-W2.V•• 


^,„,  ,,  „,,v  he  awmud  thai  nearlv  the  uh-.le  <,f  tin 
lU-neetheonre^.v-nduiL'   Ins.  nt  head  is 

-w/ ,  ^Y'  ^-(^  .yy  =/:_. 


s  i^  v.asted. 


A, 


-:,=  :';;:/.+  J-'. ^   +  J--''^.C- 


.,,,,„,  ,,„-ve.  and  elbMUs  mav  he  expre.^ed  n,  the  torn. 


/    ^  /'        al>.M' 


II,   Art.    14)- 


loss  .,fhea.Hn  the  pa...,e  ..r  the  .ater  thn.n.h  the  port.  n,av 


be  expressed  \u  tin'  lnrn>  /'^,^^- 

•1-h,    l„ssn.cas,nned    by  Calves   niayls..   he  express, 


jd    h>- 


/■"-• 


'I'luis  tiic  total  1( 


HYf^K.-irrK:   Hh'-lKE. 


./■+/■'''. -.^ 


^.K- 


Tli 


e  coclticicnt  f      m;iy 


bo  <'i\-t.n  ,iiu  (U'^ircd  \alin-  lutwi-rn 


o  and    y-    1)>-  tuiiiini;   ;i   valve,  so  that  any  cxccs-^   ••!   j.il-- 
nia\-  he  (k-~ti<)yc(l  and  the  s])ced  rc^nilated  at  will. 

I  /       ///,•  /.,'ss  /.„  .//,<•  /•'  the  Velocity  wttli  '.•hiJi   l-'u    W, 
haViS  till   /)i.u/itiixi-/'//'i  . 


i»  TCT 


/■.    =  ...  ■/■:,.■ 


II.MK 


the  ctTcclivo  head 


A, 


1/    -:    /    a-  /    -;■  /.   4-  /     1-  /-  V 


and  tlu-  I  rt'iiu  :k>'         I    '  -  ',  -  /.,  J-  /.,  -f  /..,  +  /.,  +  ''•,•,  -4-  /-J- 

6.  Brakes.  livdraulic  rcsi-taiuo'^  ah-^orh  iiht;.;\'  wliitli  1-; 
piMponionil  V<  tlic  -cjuarc  of  ti'r  -in-cd.  'I  lii-  iiroprrtv  lias 
l)efn  takt-n  .id\ani.ii^c  of  in  thr  (k-;iL;ii  "(  livdi  au'.n.  hr.ikt  -  tor 
an-t.-tiiv_r  the  motion  o'"  .i  rapiiUy  nuAinL;  mas-,  a-  a  'j^wn  or  a 
tram,  ofuci-lu    "'.       In    li;.;.   -""   thr  tl',:id   i-   .aUowcd  to   p.i-s 


"T 


Irnni  one  sid<-  of  the  pi-ton  lo  th<-  other  throu-h  .irific.s  in  the 
pi-'ton. 

l.et  w  In-  the  r.Ui"  of  tli(  ana  o|  the  pi-tiin  to  tlu  (  I'lntive 
area  ot'  the  1  M'itices. 

Let  ;■  be    the  vehuitv-  of  the    jii'-ton   wIh'M    movin-   under    a 

lorre  />. 

Let  .1     be  the  seitioiial  .ire.i  ot"  the  e_\  linder. 


*  it 


1 


-  i' 


MICROCOPY    RESOLUTION    TEST    CHART 
ANSI  ond  ISO  TEST  CHART  No    ? 


1.0 


I.I 


?  R      SJil  7  5 

'      1111=21 
2.0 


illll[ 


1.8 


1.25    ill  1.4 


1.6 


^      -APPLIED  IK/MGE     Inc 


354 


HY URAL  Lie  HKAKH. 


Thon 
tlic  work  (iiiiii-  nor  sccniid  =  /'."■ 


the  kinetic  eneri;\' ]iri  ulucei! 
( ;//  —   I  i-r- 
2  J,'        • 


=  rv'./; 


aiiil  therefore 


/'  — Tc./fw        n- 


whiJi  i-  the  force  re(|uire.l  to  overcome  the  h\(hauHc  resistance 

,it  tlie  s|)ee(i  .'  . 

Let    /  ■  lie  tlie  initial  \  ,ihie  of  ;  .  ami  /',  the  niaxinuiin  \alue 

..f  /■       'ih.  ■• 

J'' 

r         r  ■.  \^»i  -      I  r      . 


I 


i! 


ts:« 


I.i-t   /•  We  tiu:  friction  ot  tile  slide.       '1  lun 


/•      r  —  xivhiii  -  I)' 


2.;'^ 


/■ 


aii.l  /',  :  /  is  the  ina\iimiii\  retarding'  force.  It  would  cer- 
t.iinlv  I  e  an  advantage  if  liie  retarding  force  toiild  l)e  constant. 
In  ..rder  that  thi>  ini-lit  !»e  the  ease  O// -  I  !.- nuist  i)e  con- 
stant, anil  therefire  as  .-  iliniinislies  /;/  shoidd  increase  and 
consr.iiiently  the  orifice  area  diiuinisli.  X'arious  lievices  have 
lireti  adopted  to  produce  tills  result. 

.\ssuniin;4  the  retanlin:^  force  to  I'u  constant,  let  i  he  tlic 
piston's  .iistanci-  from  the  end  of  the  stroke  when  its  veloeil>- 
i-  'I'llell 


and  tilt  lefore  ,-'  is  proportion.il  to  .r. 
Hut  ui/       I  .    !•<  ci'Hstanl. 
Tlicrefore  i  w        I  >  is  iiuersele  prop.irtional  to    ♦  i. 


EXAMPLES. 


355 


ll 


EXAMPLES 


I.  A  4-ton  hylranlir  jack  with  a  2-iii.  ram  ami  a  l-iri.  jilimL;er  is  to 
lilt  a  \vcij4hl  of  I  ton,  and  is  workfrt  by  a  handle  with  a  levcraj;c  of  12 
111  I.  If  tlie  cfficifiKv  of  tlic  lack  is  80  per  cent,  wliat  force  must  be 
applied  to  tlie  I'.and'r  ■'  .Ins.  5:;,'j  lbs. 

2  Tlie  ram  of  an  luiiiaulic  press  has  a  sectional  area  50  times  as 
jjreai  as  the  pump-plunder.  The  meciianical  a'vantafjc  of  ilic  lever  is 
10  to  I.  If  a  force  of  50  lbs.  is  exerted  on  the  han-lli'.  lind  the  |)ressiire 
on  tlie  rain.  ./■'/i.   2  5.<XX5  lbs, 

V    A  fori  e  of  /'  lbs.  is  reciuircd  to  puii(  h  a  hole  of  </ ms.  ciiatiieter. 
I'ind  liie  diameter  of  tlie  ram.  the  availaljle  fluid  pressiiie  beinsj  /<  lbs 
per   stpiare    inch.     If   tliis   pressure   is  developed   liv  a  steain-intensilier 
with  a  steam  piston  area   11  timesih.it   of  thi-   iniensifier's  r.mi.  llnd   the 
required  steam-pressure.  ,^  /■ 


./;/ 


\ 


.    P 
11/'    It' 

.\.  In  a  sterl  hvdraulic  press  the  fluid  pressure  is  Oocxj  lbs.  per  square 
incli,  and  the  ina.ximum  allowable  stress  in  the  metal  is  iS.ooo  lbs.  per 
sipiaie  inch.  If  tile  internal  diameter  of  the  press  is  12  ins.,  what  must 
the  tiiicktiess  of  the  metal  be?  If  the  thickness  of  the  riietul  is  3  ins., 
what  must  the  internal  diameter  be .'  Aiis.   2.4K5ins.;   14.4X5  ins. 

5.  A  straif;ht-line  law  is  fuuiid  e.xpi  limentaliy  to  c<ii;:iert  the  wei>;lu 
If  to  be  lifted  and  the  etTort  A"  on  the  h.indle.  Kind  the  l.iw  from  the 
loUowiu};  data;  when  //=  1O05  lbs..  A'—  10  lbs.,  and  when  //■=:68o5 
lbs..  A"  =  50  lbs.  A  pressiireyauye  i;iws  the  flipd  piessure  as  ly3J  lbs. 
per  square  inch,  when  /f  =  7ooo  lbs.,  lind  the  fiictional  loss  at  the 
le,itlier.  anil  if  there  is  the  s.inie  percentage  "(  I"ss  i.t  the  two  leather<4 
lind  the  law  connecting  I',  .mil  the  foice  /'  on  the  pliin^jer.  The  experi- 
H'ents  were  made  on  a  j.uk  with  a  ;1  in.  ram,  a  \-\\\.  plunfjer,  and  a 
lever  with  a  velocity  r.iiio  of  v>.     ( Perry.  1 

Ans.   If"  —  30:;  +  I  ',0  E:  9.1  per  cent :  /'  =  41  +  17.5  /•.". 
il'erry's  "Applied  Meclianii  s.") 

6.  \\\  .icciimiil.itoi-r.im  1-  1."  in^.  in  diameter  and  has  a  stroke  of 
.'1  ft.  Find  the  store  of  enei(;y  in  footpounds  when  the  ram  i^  at  the 
l'i|i  of  its  stroke  and  i«  luadrd  till  ihe  pressure  is  750  lh>>.  per  square 
inch.  Ahs.  •<5«.3:o  ft. -lbs. 

7.  III  a  dilfciential  accuniiil.ilor  the  diameters  of  the  spindle  are  7  ins. 
•ml  5  ins  .  the  stroke  is  10  ft.  Find  tin-  st.re  of  energy  wlien  full  and 
loaded  to  2000  lbs.  |>cr  s<]iiare  inch.  Am.  377,oc»o  ft, -lbs. 

S.  The  pressure  on  a  5-in.  ram  is  ti-  !■•    \---j  lbs.  (a-r  square  inch,  and 


ilh! 


t:-^ 


%, 


y,^> 


i-x.-iMi'irs. 


lliL-  .siip.ily  ("incs  from  a  t.iuk  ioj  f: 
ot  the  pi.ilDii  ill  llie  iiitciisiticr. 


ii.    riiul  tlif  m-ci-ss.iry  <li;ir"etcr 
.Ills.  ^4  ins. 

9.  InaililTercntial  press  t lie  ilianK-UTs  ..J  iln'  upper  .ithI  lower  porlions 
of  the  ram  are  6  ins.  and  cS  ins.  rcspeclively.  'I'lie  pres.sure  is  1000  lbs. 
per  stpiare  inch,  and  tiie  stroke  i.s  10  ft.  Find  llie  lo.id  on  tlie  accumu- 
lator, the  maximum  si.  ire  of  energy,  and  the  store  of  water. 

.///t.  22,00c  lbs.;  2JO,txx}  ft.-lbs. ;   iJJ  cii.  ft. 

10.  What  load  must  be  applied  to  a  dilTerential  accumulator  to  (,'ive 
a  pressure  of  i<)oo  lbs.  per  scpiare  inch  ?  Tiic  upper  and  lower  diame- 
ters of  the  r.im  are  3  anil  33  ms.  respt ctively.and  tlie  friilion  of  the  cup- 
leathcis  m.iy  be  taken  .is  5  per  cent  .f  the  gross  loail. 

yliis.  to62  lbs.  ;  (.700  lbs. 

11.  Find  t  ;e  weight  which  will  give  an  average  fluid  jire^sure  o: 
7,0  lbs.  per  SI  .re  inch  in  an  accumulator  with  a  14-in.  ram  and  a 
stroke  of  10  ft.  How  mudi  energy  can  be  stored  up  ?  Find  the  friction 
at  each  cup-leatiier,  assuming  tliat  between  .slow  rising  and  falling  the 
pressure  fluctuates  between  7S0  and  73S  lt)s.  per  squ.ire  inch,  ll  the 
pressure  is  750  lbs.  i)er  squ.ire  incli  at  mid-lift,  find  the  actual  fluctua- 
tion. .I'll.  1 1  5,500  lbs. ;   i,848,o(K>  ft.-lbs. ;  3234  lbs.  ;  37O9  lbs. 

12.  .\ii  accumid.itor,  loaded  to  a  pressure  of  750  lbs.  (.er  s<|uare  ini  h, 
has  a  ram  of  21  ins.  diimeter.  with  a  stroke  of  24  ft.  How  much  H.P. 
can  be  obtained  for  a  period  of  50  seconds  ?  ./«...  226.S. 

IV  .\n  accumulator  under  a  loatl  of  200,000  lt)s.  is  to  transmit  10.) 
H.I',  througii  a4-in.  pi|>e  I  .nile  long  with  a  loss  of  10  percent.  Wli.it 
shoul  i  lie  the  iliaimter  of  the  r.un,  the  coetFlcient  of  pipe  friction  being 

.ooo  }  ■'"■*•  'r-.v?  ''■•'• 

14.  .\  stiam-aecumul.iti  r  h,!s  to  develop  a  total  force  of  6O.000  lbs. 
upon  tlie  r.im  of  a  puiuii.  Tlie  piston  area  is  15  times  that  of  the  liy- 
draulii-cyliiidei,  wliicli  hisa  diameter  of  10  inches.  Find  the  intensities 
of  the  steim  ,ind  the  w.iler-piessure.  .hn.  56 lbs.  ;  S40  lbs. 

ij.  The  pi-.ton  and  r.im  areas  i.f  a  ste.im-accuniulator  are  in  the 
ratio  of  10  to  I.  Find  their  diameters  so  th.it  a  sleani-pressure  of  100 
lbs.  per  sq.  in.  tniy  develop  a  total  ioid  on  the  ram  of  38,500  lbs. 

.Ills.  22.136  in.s, ;  7  ins. 

ifi.  .\  Mrotheihood  engine  with  a  4-iii.  cylinder  and  a  3-iii.  stroke 
makes  50  revols.  jh-T  minute.  The  average  motive  pressure  is  ;i«  Ib- 
iMT  so.  in.,  and  the  avi-r.tge  b.ick  pressure,  due  to  frictional  lesisiames, 
etc..  is  210  lbs.  |)er  s'|.  imh  Find  tiie  II. P.  ilevelo|K-i|.  and  also  deler- 
mine  the  diameter  of  the  lylinder  if  only  <'iu-  //<»// of  this  power  is  to  be 
developed.  -'««■  7:  ^^3  •'"*• 

I".  .\  crane  with  an  hydraulic  elTicienry  of  .0  and  ,1  met  hanical 
ertiiieiicy  "f  ,45  is  worked  bv  water  at  a  pressure  of  750  lbs.  per  si). 
inch.  The  piston  has  an  effective  area  of  gf^  »q.  ins.  on  one  side,  48 
sq.  ins.  on  the  other,  and  pushes  a  three-she. ivc  pullcv-hloi  k.  Find  the 
m.iXiinum  >»'i    bt  wbi.  b  .■  m   'm    lifti-d  .iiid   the  Work   done   i-er     alloi  of 


ix.-iMri.i:s. 


),S7 


1 

i 


viiCL/hs/  wlicn  the  \v;iu-r  presses  on  one  side  only,  and  ser.^>i</ whi-w  it 
presses  on  both  sides.  Also  lind  tlu-  work  don.-  per  i;.illo'i  of  w.acr 
wi'.L'u  the  lull  loads  in  the  iwo  kinds  nl  \vurkin<;  are  beins;  lifti<l. 

,1>!S.  4S()o  lbs,;  699^.4  tt.-ll)s.;  343^  ^^>^--  iWJ--  H.-lbs.;  699.S.4 

ft, -lbs, 

iS,  An   iiydrauhc  eran,-   with  a  velocity  ratio  of  9  and  a  niurhanual 

efficiency  of  .75  bas  to  lilt  a  wti^;lit  of  10.000  lbs.     It  is  worked  by  water 

at  a  prcssuio  ot  750  lbs.  per  s(|.  in,,  and  the   frirtional  loss  of  pressure  is 

91  lbs.  per  sq.  inch.     Find  the  dianieier  of  the  ram,  A>is.   15.2  ins. 

19,  The  two  wire  ropes  Ironi  the  cai;.'  "f  .'  lani-lift  pass  vertically 
over  a  pulley  to  aconnterweigiit.  anil  the  1,1m  n-es  Iroiii  100  ft.  below  to 
20  ft.  above  the  level  of  the  suppl\-pipe.  Water-pressures  01  500  ibs. 
and  100  lbs.  per  sq.  in,  art  upon  a  3i-in,  and  a  7-111.  ram,  respect nely. 
Find  the  weiubt  of  the  ropes  per  lineal  toot  ,ind  the  lilliiii,'  lorce  at  the 
toi>  and  bottom  of  the  >troke. 

.■l>is.  4.2  lbs,,  16.7  Ids,;   5230  lbs.,  4729  Ibs.  ;   5521  lbs..  3516  lbs. 

20,  Find  the  piessute  due  to  mertia  at  tne  end  ut  the  out-stroke  of  a 
rotary  motor  with  a  4-m.  I"-ton  and  a  7-1.1,  stroke,  driven  by  water  in  a 
4-111.  supply  pipe  250  tt.  lon^.  The  motor  m.ikes  125  revols.  per  minute. 
,,nd  the  length  of  the  connecting-rod  is  15  inches, 

.his.  20.7  lbs.;  12.9  lbs, 
ji.  A  direct-acting  lilt  has  a  r.mi  9  inches  diameter,  and  works  under 
;i  ,.'//v/,/«/  head  of  73  leet.  of  winch  13  per  cent  is  rerpiired  by  ram  liic- 
tion  and  tiiction  of  mechanism.  The  supply-pipe  is  loo  feet  long  and  4 
inches  di.iineler.  Find  the  speed  of  steady  motion  when  raisinj;  a  load 
.,1    1350  lbs.,  and  also   the  lo.id    it   would   r;ii>e   at  double   that   speed. 

If  a  vaive  m  the  supply-pipe  is  partially  closed  so  as  to  increase  the 
.oelliiicnt  of  rcMstance  by  ,'.  what  would  the  speed  be? 

.Ih.s.  .-^peed  =  2  ft,  per  second  ;   lo.ul  =  I  50  lbs, 

22.  i:i"ht  cwt,  of  ore  is  to  lie  rai-ed  liom  a  mine  at  the  rale  of  9<xi 
feet  per  ininute  by  a  water-pressure  euKine,  which  has  tour  sinyle-actin-; 
cylmders.  6  inches  diameter,  18  inches  stroke,  making  ('O  revolutions 
per  minute.  Find  the  diameter  of  a  supply-pipe  230  feet  lon^{  lor  a  head 
of  230  feel,  disre^atilinj;  tesistances  and  taking/  =  ■<>"'i- 

.his.  Diameiei  =  4  inches. 

2V  if  \  be  the  length  ripiivalent  to  the  inertia  of  a  water-pressure 
enpiV.-,  /■  the  coelhcicnt  of  hydraulic  lesisianco,  both  rcduce<i  to  the 
r.ini,  7 .  the  speed  of  stea.ly  motion,  lind  the  \eloeity  of  ram  after  moving 
from  lest  tliroiiKli  .1  space  ..  a){ainsl  a  constant  useful  resistance.  Also 
find  the  tim< npini. 


f 


.hly 


I    -   f 


\ 


'   * «  '0         * 

24,   An  hydraulic  motor  is  diiven  from  .111  .iccumiilator,  the  piessure 


.55S 


f.\.-im/uj:s. 


in  uiiii :li  is  750  lbs.  per  s()uarr  inch,  liy  means  I'f  a  Mi]ipiv-pi;H-  yoo  lc<.t 
luni;.  4  melius  (iianictcr  ,  u  hat  wnuiil  be  llie  ni,i>inuini  pDwer  uieureti- 
caily  .tilainabic.  an.l  wliat  would  be  tlio  velocity  in  llie  ("pc  correspoini- 
nij^  lo  tlhi'.  powii  .-  i'lnd  aiiproximalcly  llie  etliciency  of  transmission 
at  half  [louer.y  --  .007. 

W«j-.  H.l*.  —  250;  :■  —  Z2  It.;  eiruleiiLy  =  .66  nearly. 

:5.  .\  ;.;un  ici  (jils  uiili  ,i  ni.ixmuini  velocity  of  10  (eel  per  second. 
The  area  nl  the  o;  ilice-i  in  liie  (_onuMe.ssor,  after  allowing  for  coMtracti(Jii, 
niay  be  taken  as  one  tweiitietli  tlie  area  of  the  piston.  I'lad  the  iiiiti.il 
pressuie  in  tue  coInllres^o^  in  feci  1  if  ii(iui(l. 

.AssuMiing  the  weight  i.f  the  gun  to  be  12  ton^.  Irulion  uf  slide  3 
tons,  thanicter  I'l  coinprr-sir  o  iiu  lies,  lluid  in  c  iniprosor  water.  Iiiid 
the  recoil. 

I'inii  the  mean  resistance  lo  recoil.  Conipate  tlie  ni.i.^iinuin  .ind 
mean   1  r^i^laiucs.  r.icli  i-.Xl  his,  Vi-  nl  liuiimi  uf  >hdr. 

.i/ii.   oji  ,  .;  11.  jj  111.  ;  total  mean  lesi stance  --  4  4  Imiis  ;  latio  —  J.  5. 


iiAr'n-.K   V 


IMl'ACT,   KKACTlnN.   IMl'X* 


i     AMI     TASl.KNTlAL 


•rrKi!iNi> 


Noll. 


I'lu'  IdHnwin:-;  '-nihImm'^  arc  u-ol 


=  the  vcl"Lit\- 


)f  the  ict  hcfnrc  imi).Kt ; 


atlcf  k-a 


viil'J  tlu'  vane 


\atli 


"     •■    water  rcl.iti\cl\-  ti-  the  vane: 


It  = 

ir .        ,1  II 

A  —  sectional  area  of  the  iniimi-ini;  jet; 
\n  =  nia-^s  of  the  water  rcachnv.;  the  vane  per  -eeona, 
I    Impact  of   a  Jet  upon  a   Flat    Vane   Oblique  to  the 
Direction  of  the  Jet.    Let  ^*  be  the  anc^le  between  the  norn.al 
t,,  tlie  vane  an-!  the  direction  of  the  ini,.ini;in-  let,   '/'  the  an,Me 
1),  tween  the  normal  to 
the  \ane  and  the  <hree- 
ti,.n  ol"  the  xane--  mo- 
tion, and   (1    the   aiv^le 
lutween   the    \ane  and 

tlie  vertiial 

Tin-     let,     niovniL,' 

with     it-     stre.un-uncs 

]Mr,dlel,       -xvelU      out 

near     the     vane,    o\er 

which    it    -.prcad.   and 

with    which    it    tr.i^els 

iiloiii,'  in  the  iliicction 

„f  the   xane's   motion,   and   hn,dl^    a„,nn 

stn-an,-lines  sensibly  iwrallcl  to  the  v.mc. 


>    .v 


/ 


/ 


\^^>    ~-~^^C 


^-"^ 

\       ^    >^ 

'  V 

\    '  ^ 

\ 

;r 

\ 

/ 

f 

/' 

t' 

Fi. 

:.  20I. 

llows  aloii^j   witlj  its 


|l 


3?9 


360 


IMP.-tCT  OS  1 1. .IT   l-.-iSF.S. 


I 


^m- 


The  prnhK-iii  1-^  >till  t'urtlur  cirii[)li(.-,itiil  hy  tin  pnnlLicti' U' 
'•t  ijil'vlios  and  \iirticc-^  icr  which  .illnw  aiK  c  i.;i!i  Mni\  In-  inadi,; 
in  a  [jiirclv  (jiiipirical  iiiamici. 

Let  X  hi-'  the  normal  J)^l■-^^ll!•l■  mi  the  \,nu-  ihic  to  the 
inipat  t. 

l.i-'t  -\      111-  the  tiital  i!i>inial  ])n."^^iirc  i<u  the  \ani". 

I.i't    iriie  the  \\ci!4ht  lit   .\atir  i.n  the  \ane. 

■Iheil 

.\   =:  .\  '  —   U    M\\    (I   -.  chan,L;e   <it    niMnientimi    in   iHrectiDii   of 

the  IK  11  nial 

■=z   iiir^   Cos    H    —    ;////  Cos   0^ 

or 

A  /'/(."■,    cos    H  —    //  COS    (/)|,  .         .         .        ('l  ) 

(N.l!.  '1  he  si.Mi  in  li'ont  of  u  cos  <,'>  will  he  [iliis  if  the  jet 
and  \ane  nioxc  ni  o]i]>.,sJte  (hiection-^.) 

1  hi  term  /I'sin  <i  nia_\-  In;  desij^iiated  t!ie  .\7////.-  pressure, 
and  the  teini  iiiw^  cos  l>  —  //  cos  f/) )  tlie  ifviiiiiitit  pressiin-, 
wliieh  causes  tiie  ileviatioii  ot  the  slieani-hnes. 

Xcili:. — 'I'lie  ])ressnre  when  a  jety/'/.v/  sfikes  tlu'  pi. me  is 
^'renter  than  when  tlie  llow  iias  liec<Miie  stead}-,  oi-  a  pi  rm.ment 
rej^iuie  i--  estahlislied 

This  is  ni.ade  es'idi  nt  h\-  the  to]  1,  ,\\  in-  i  oii-idei.ition  : 

At  .m\  moment  h  t  MX,  /'(}.  A'.S  he  the  houndin;,;  ])l,me- 
across  which  the  water  is  tlowin-  with  its  stream-lines  seiisihK 
])aralle!. 

In  a  unit  of  time  hi  the  liounilin;^  piaiu's  of  iju-  mass  he 
.1/  .\",  /•  c/.  A'  .s'. 

I'lien,  itiitially,  tlie  reactiois  of  the  ])lane  must  destroy  tlie 
inotioTi  of  tlie  mass  of  the  fluid  hounded  li\  .1/  .\'  .  /'(Y,  and 
A'.S. 

'lake  ('("  to  re])resent  Tj  in  direction  and  maL;tiitude. 
•  •     ( '/:   •  •  •  •  // 

In  one  second  the  vane  A/i  moves  parallel  to  itself  into  the 
position  .r/>".      Let  .I'Ji'  intersect  ( '( '  in  J). 


I 

i. 


ThL-n 


IMPACT  OS  FLAT   t^ANES. 


VI  =  -A  .  DC  ^      Ai7;  —  (V?i 


COS  0 


301 


Tluis  equation     1     liccniiu 


A' =  {:■   cos  H  -    II  COS  0)- 

i,""  cos  w     I  '  ' 


(2) 


(3) 


I. ft  /'  hr  till.'  i.|-c--^urc  ill  tilt.'  (lirrctinii  dftlu'  wmc's  niotioii, 
Tlicii 

"''1  '-"^  ''' 

/      —    .\    Cos   </)    :^         .] ; C     Cos    H   —   II   COS   0)'',     .         (J.) 

<;•      cos  W      '  '  ^• 

am!  tlic  iisi/i</  wcrk  done  mi  tlic  \anc  per  --ceciiiil 


:c     Cos  <l> 

-   Ill   '-=        A  '  .11' '\   cos    H         11  Cos   01-. 

j^       Cos   " 


Tlic  A'/(f/  ixvdiliihlc  xcork  =-      A—-. 


■     (5) 
.     (6) 


•i'     cos  0    . 

.1   ,."t"',    cos  //  —  //   COS   01- 

1 1  »i        ,j-   •  i,''      COS  «    ■   1 

lli'lHr    the   iff/i'/r//iV    --   -^  . 


COS  0    ti 


'riiis  is  a  m.ixiiiium  wIut. 

7',   COS    W    IT    7,1/   COS    0, 


ami  tlicrcfoii 


tlu'  max 


S 


iniimi  ((In.  u-iu  \ 


cos-   f1. 


(S, 


(9) 


it  tlic\.mc  is  1))  small  sccticmal  arra.a  pn-tion  <  if  the  as  .itrr 
uill  csca])c  o\cr  the  lioimilary  aivl  the  |ircssiir<  .iiu-^t  nci.(_--'-anl\- 
be  less  than  that  L;i\cii  b\-  equation  (^). 


Hi.  J  i : t 


362 


IMPACT  OS  hi. IT  r.t\i;s. 


Srrirs  (>f  I'aiii.s. — In-tcui  <A  "iic  v.mc  itiii\in,L;  bctorc  the 
jet,  let  a  -^erie^  <>!  \ane'-  l.<e  iiitii  uliii  eii  at  shurt  inteiAals  at  the 
'^anit/  iiniiit  m  tin-  path  ^f  the  jet. 

Iht  (luaiititv  ol  water  now  reachiili;  tlic  vane  [ler  -ecoiid 
is  e\  iilently 


.s 

and,  b\- e(iuatinii   j),  r!ie  normal  ])res:"-iire 


(lO) 


A'=      ./;■,(; 


./ryr',  cos  t^  —  u  cos  0). 


(>'. 


Also,  the  /'JcSS/t/<   in  the  direction  d  the  motion  of  the  \ane 
-  /'  ^  .y  cos  </'  :--      ,,-/r'|(r',  cos  m  —  u  cos  <pj  cos  <p.         ^^l2) 
The  useful  Xi'ork  done  per  second 


;>  * 


'Hj 


:=  Pit  —         A'i\U\7'^  COS    «  —    «  COS   (p)  COS   0, 


and  the  cjj'utttny 


w 


~-A'\u{7\  cos  W  —  ;/  cos  0)  cos  0 


2«(7',  COS   fi   —    U  COS   0)  COS   0 


Tliis  is  a  nia.Nimuni  \\  lien  i^  Cos  »  —  2tt  co^  0,    . 
and  tlieicforc 


the  ina\inMiin  (  I'tu  ii  lu  \ 


cos-  « 


I '3: 


('4 


1 ; 


16: 


i;x.-i.\WLts. 


;,'•.) 


Ex.    1.     Let  a  single  vane    be  at  riyht  anylcs  t...  and   movt-  in  tlic 
line  'if,  ihe  jet's  motion.  Fig.  ;o2. 
Then  0=0  —  t!>.      Hence 


tin-  iiressure   =/'  -=  A'  =       ./U' 


to' 


( ir  I 


ihc  useful  work         /'«        -. /;<(?•.  -  «('; 


fH,.    202. 


ll'     till'  <'ll'i(  leni  V 

the  iiia.ximuui  elfu  leucy 


C^"-"^' 


A^.tin.  if  «  -^  o,  i.e..  if  the  vane  he  U.xed.  and  if  //  iie  tii.'  liead  ..iire- 
sponiliiig  t.i  the  velocity  f, .  then,  hy  <<iu,H,on  (ry), 

/'  =  "'  .i;',"  =  27/^7// 

=  /'U'lcc    the    weight  <if  a    (oUimn    of    water 
(if  lieigiit  //  and  sectional  are.i  -•/. 
Ex.    2.  Let  each  of  a  series  of  vanes  In-  .it  tight  angles  to,  ami   move 
in  the  line  of,  the  jet's  motion  .it  the  mst.int  of  inijiact. 
Then  Cr  —  o  —  <;&.     Hence 


the  pressure  =  \  =  /'  =  '''-^lr,iv,  -  u); 


tlie  useful  work         /'k 
th<'  elhi  inirv 


A-.'.u^'' 


2«(7'.— ») 


11  y. 


121) 
'22) 
IJJl 


? 


tilt-'  inaximuni  (*irn_ierny  —  - '24) 

Ex.3.     .A  stre.ini  (;f  .125  '^q.  f' .  sect  1011. il  arc.i  ilehvcrs  I'Mii.  ft.  of  water 

per  second  and  impinges  norm. illy  .ig.unst  a  Hat  vane,      ll  is  r<-iiuired  to 

lind  l.'l  llie  pressnie  on  the  vane  if  ti.xed  ;  (/i  the  pre-surr  ,ind  the  useful 

elfeit   ll    the  v.ine   moves  m   the   direction    ot    tb.e   jets   motion   with    .1 

velocity  of  40  ft.  per  second  ;  (n  the  pressure  and  useful  ellect  when  the 

s;ngle  vane   in   (l>)   is  replaced   liy  .1  series  of  vanes  which   follow  each 

■  itlier  at  intervals  of  a  second. 

10         ..     , 
The  velocitv  of  the  jet  l.iefore  impact  =         ,   "  00  it.  per.  sec. 

(.n  The  pressure  on  v.ine  ---  momentuni  of  |et   -       ~  x  10  X  ho=  I  5C2J  lbs. 
(f>)  The  ipi.intity  "f  w.it.r  re.u  liiiii;  the  vane  pel   sec. 
=   -(So  -  40;  =  5  cu.  ft. 


,i<'4 


IM/'.-ICT   O.V   .srRI-.-ICFS   (11-    Rl-yOI  ITION. 


It 


%¥. 


.'  » 


.;vJ6- 


t^t 


M, 


Till-  pressure  dm  ilie  vane  =  iiioinetUiiin  nl  jet 
Tlic  useful  elTeil  =  3904  x  40  =  15/125  ft.-li)s. 

><2\  So' 

Tlie  tipt  tl  ;iv.nl.il)le  wmk  =     -"  10  .  -       =  C>2.',oo  ft. -lbs. 

32  ^ 


I5f.2 


The  ctrii-ieiicy  = 


rherefoie  the  elii<  leiicy  =  =     . 

03500       4 

(<■)  The  (luantity  of  water  now  rearliiiii;  the  vaiie  per  secnnd 

=  -  X  tJo  —  10  cu.  It. 

The  pressure  011  the  vane  =  niomentuni  of  jet 

(■,21 
—     "■'loiiSo  —  40)  —  7X1 J  lbs. 

The  useful  elTect  =  781 1  x  4 j  =  31. -50  ft. -lbs. 
3 1  2  50  __  1 

025Ov>  ~  2  ' 

\'.\.  4.  The  jet  in  tlie  prcceriing  example  iuiphii;cs  upon  :;  vane  with 
its  nc'iuial  inrline<l  at  60°  t'l  lire  jet's  (iirection.  and  is  driven  with  a 
velocitv  of  ij  ft.  per  second  in  a  direciioii  making;  an  atiyle  of  30°  With 
the  vane's  normal.  ITiul  Ut)  the  pressure  on  the  vane  ;  (/;  tlie  useful 
ellec  t. 
(.M  The  rjuantitv  of  water  reaching'  the  jit  pir  second 

The  relative  velocity  in  the  direction  ol  the  n  irtnal 

=  80  cos  60"  —  20  cos  30°  :-  10(4  —131       j:!.^8  ft.  p<'r  sec. 

Tlie  normal   piessuie   np<in   tiie  vane;  =  uionientum    in    direction   nf 

normal 
=  5.67  X  22.68  =  i2S.()  lbs. 
The  pressure  in  direction  of  v.uies  motion  =  i^S.ft  cos  30'' 

-  111.35  'bs. 

{!>)  The  u.seful  effect  -_-   MI. 35  X   2o  =  2227  ft.-liis. 


cos  3o<''\        ; 
80-20         ;        =     (4  -  t 
cos  00°/         2 


31  =  5.''i7  cu.  ft. 


The  etruiencv 


'i2;ooo 


.035^1. 


5.  Jet  of  Water  Impin^nni^  upon  a  Surface  of  Revolution 
Moving  in  the  Direction  of  its  Axis  and  also  in  the  Line  of 
the  Jet's  Motion. —  1  he  rL-l.itivo  \el>>cit\-  ot  the  jet  IS  r,  -  .. 
if  .lu'  \c\  an  1  -urfuc  nvne  in  t!ie  .same  tUrection,  I'"ii:^s.  203 
aii.l  204,  ami  v^  ~\-  11  if  llie}'  move  in  opposite  ilircctions,   I'iys. 


.:^iJ 


IMP 


ACT  OS  SURhACFS  Oh  REl^'OIHTION, 


j"3 


:o;  ani 


1  ;or).       Tlii^  relative  vc 


locitv,  it  friction  is  ilisrc^sirdod. 


remains  iinchan;^H-,l   in  mai 


itui' '  as  tilt!  water   flows   over  tht 


su 


rface,  but   the   stream-line  direction  is  do 


Icviatetl  throuLdi  ai 


antile  fi. 


^~<t....ii 


^i.^.-V--. 


I'll..  204. 


f!^c^s:-^-!^„ 


'~^--':". 


.  .J) 


F         " 
>. 


Fii;.  2i)f). 

Let  ill.-  watrr  K'a\<'  tile  surface  at  /'.  and  in  tin-  direction 
,,f  tin-  tanL,'ent  at  P  take  Pli  --  :\  n.  I'i.L;'-  -";.  and  J04, 
and  /)/:'  -^  .-,  -t  //.  I'i:-;-^-  -u?  ;'">'  -"^'-  '^'■•^^'  /V  parallel 
to  the  a\i->  and  take  /'/'  --  it. 

ConipUte  the  i)aralleloL;rani  PiJ.l  - 

Then  Pi'.  the  <liaL;onal.  inu^l  represent,  in  direction  and 
lua-nitude,  th.e  absolute  \elocit\-.  v.,  with  which  the  water 
leaves  the  surface. 

Hence,  frt>ni  l-'i-s.   J03  and  J04, 


w-   i-  1; 


uv 


inr, 


,()  cos  I  180="  -  H),    .      (l) 


and  the  work  done  b\-  tlu.'  water  on  the'  surface 


=  in  11^ ." 


uv  I  —  COS  (i) 


wA  ,    .      1^ 

>       u  V       ur  sin- 
g  2 


(2) 


if 


3^'^'  /U,"  •/(,/■   (),V   ,s/  A7,-/C/:'\    Of    h'f  r<)l !  T/ON. 

li'.in    1- 1->.  jt)5  and  Jo6, 

'■/  =  "■'  +  ('"i  '',    "'■'  —  2/<if|  -I-  /n  CDS  /i, 
and  tlu'  '.vofk  dune  hy  tin-  snrtacr  on  tin-  water 


/// 


wA  /y 

2        u  V        u     sin 

«  2  •        •        • 

Lot  /'  lie  till-  pleasure  on  the  surface  in  the   dir.rtion 
Mlntiun.         I'llrn 

n  ,      .  wA  /i 

irU        wiii'K  dorii'        j  u*  V     •    u      Sin' 

g  2' 


13) 


>t  Its 


.'I lid  tlicrcfun- 


„         wA  ,      fi 

r       2         V      11     sin 

g  2 


1  lu-  if//i  ii /,:;   tipf  tJK'  lasi'  1 1)  I'lj^s.  jn^  and  2 


04 


?       //.,-■,  —  H)-  sm-^        4//r,-,  —  //j-  sin'^ 


U) 


/  .■• 


j,« 


(5) 


itat 


t&^ 


\\  Im  li  IS  a  iiiaMiiuM't  and  _  sin'       when    :•      -  yi. 


Sifiis  <>/ Siirfiui  s.  —  U  a  nimdnT  <>("  surfaces  an  -m  i.  s>im  ly 
i  :tn.d„cf(l  at  slmrt  intervals  at  tlie  same  point  in  the  jiath  of 
the  jet.  the  iiuaiitity  of  water  reaching,'  cacli  surfaco  jh-T  scc<>nil 
luconu's 


M  = 


u'A 


'  f 


In  till-  ease 

wA  .    ,  ^  -r,, 

the  work  done       2  „  v.u  v,   :   u    sm      .    •      •      '") 

g 


wA 


/^ 


and  the  pressure       2       v,iv,   .  n    sin  ^ 
Also,  the  efficieiu>-,   wlun  tli.'  n.U.-r  .Invc-  the  siirtacc, 


•7) 


i.'J 


,/i 


TV,;r,  -   /.    MU-^ 


T>'.  I   ,- 


4«rT',  —  w)  sill- 


/i 


-.3 


(8) 


which  is  a  inuxinuim  ami         sin-  ^   when.,        211- 

With    a    er,Mvex     Mula..       /i  <  <)0  .    an.l     tlu-     (MeflKient 

-.  ^j,,-'  ''    ,„•   I   _  cos     ,  is  less  than  unit>  . 

With     a    e.MK-ave     surface    fi  ,>  «>n  .    and    the    cnetTuient 

2  siir  1  or    I  -  cos  li.  is  -reater  than  unity. 

If  the  surf ..  .    !'■    "f  ill-  cup  tyiu-  an.l  luniispherical,  the 

maximum    eftKRiuy         ~ni      '_,      r_-i      ^nui-,1-=iXo'        The 

water  siiuuhl  tiierefoie  leave  the  su.l.u.    uiil.out  veh-ut'      m-! 
suhstitutm-  '"   -ul  //    -   i.Su    in  e.iuati-.n     1.. 

/,  ,J//         //)•'  —  2//(2/' 

Ex  A  i<-l  of  w.ilcr  ..(  .125  s<l.  ft.  s.rtional  area  .1.  1, vers  1;  ,u.  It.  ..( 
waUT  an.l  iniphiKes  axuilly  U|.on  a  i.'o'  e..n.-.  Fu.d  (,n  th,-  ,.n-ss.T.-  ..n 
the  cone  when  fixed,  iin.i  (*)  Hie  (.resiiire  on  tlie  cone  an<l  the  useful 


yjS 


iMi'  tcr  o\  ihiRniiKnn  r.-ixi:. 


!r  '" 


fct4 


N. 


lilfcl  uhi'M  tin'  coiic  is  fliivrii  in  tin-  (iircctidii  nf  its  axis  witli  a  vclority 
III  2-  ''    li'T  srCDiid. 


Tlic  VL-locity  "f  the  jet  tx'fotf  impact 

62.'. 


=  '/)  It.   iicr  sec. 


(./)  I'nssiirc  on  con:;  \  M.rf  nr   -^      2  -'-  .  12.96  sin'   "'      -  uj^  lbs. 

Picssiirt-  (111  (('/Kiiiv  Mil  lace  =  2  ~'t2.</>  sin'  — "      =:  ^7~c  lljs 

j:  2  JJ'3      ■• 

(fi)  When  the  watir  iin|)ini;cs  on  the  iiun'fx  sntface 

till-  work  iL.nc  =  2   '}■  ^^32(1/)  —  ',21'  sin'—  =  16.000  ft  -lbs. 
,2  is  2 

l6oo<j 

tni-  pnssnre       .-=  ^     =  500  lbs. 

When  the  water  inipinycs  on  the  .omave  surface 

the  \v.;rk  <lnne  =  2  [\'  ^.52(9^)  -  ;,2i'  sin'  —  =  4R.000  ft.-lt)s.. 


I  he  pressure 


4S000 

-       =  Ijcxj  lbs. 


6.  Impact  of  a  Jet  of  Water  upon  a  Van.-  with  Borders. 
—  l.Lt  ihc  \aiii  m  .\rt.  I  l(c  piMvuiotl  wit!,  borders,  i-'i;,rs.  207 
and  j(,),S,  so  as  t..  |)r<iclute  a  further  (!e\  i.itioii  nf  the  stream- 
lines, .md  let  tlie  water  tniall>  (low  off  witli  a  velocity  ,-.  in  a 
ilirection  makin;.,^  an  an.L^dc  "'  witli  the  iiomial  t,i  the  vanJ. 


r 


Then 
tin-  normal  pressure  i:;  A' 

■  ^  ;//.•',  cos  H  T  »/r.  co^  "'  :^  inn  cos  </> 
=  ;//tr,  cos  f  T  "'^  cos  W'  z^  ;,  cos  0i, 

the  sij;n  of  the  second  term  hein;.,'  pins  or  minus  accordiiifj  to 

'111    ilii.itiMii    ill    uhi,  I1    :]:,     ^tiriin-liiK  -    .ire    finalK'  devi.ited. 


...4 


•lb,.   HT'.t   nf  the  hordci-   i^  thn-Hnrr   f.  increase  .,r  diminish 
,1,,    „,,,nal    pressure,  ami  hence   aUo  the   useful  u-rk    and  the 

et'tkiencw 

sn  ,'i  \l.   Cam,. -Let  the  vane   he  at  rest,  i.e  ,  let  u  -  o, 

and  K-t  tlie  hn.d  and  initial  directi.m.  of  the  jet  he  parallel. 

,-'  .      Then 


Al:..),   let  ,- 


,V=  m:\  COS  .''  +  T',  t'"S  ») 
g 


—  4:<vl  //  cos 


e. 


Hence    if  «  -  C  the  iiMnnal   pre-Mire  A'  -  47e.-I//        A'/'^ 
times  the  uei-ht  of  a  column   m  uater  m    height  //  and   sec- 

"  "7  Impact  Apparatus  in  Hydraulic  Laboratory,  McGiU 
University.  -  1  hw  apiur.>tus  uas  cnr^trncted  IV.r  the  pur,.osc 
of  cleterminin.4  the  force  uith  whuh  .Ms  from  onfices.  no/./,les. 
etc      impin-c  upon  vanes  of  d.tterent  forms  and  sizes. 

\  massive  cast-iron  bracket.  Fi^'.  209.  has  one  end  securely 
bolted  to  the  front  oi  the  tank,  and  the  other  supported  l.y  .1 
vertical  tie-rod  from  one  o|  th  o.d.  heams  in  the  ceding.  1  l>c 
upper  Mirface  .s  proxid,.  d  uith  accurately  pl.mcd  shdes.  uhuh 
,re  set  level  about  ;  it.  above  the  orifice  ..xi-  If.  Tron,  anv 
cause,  the  end  of  the  bracket  farthest  from  tlu  t.ml,  1^  lound  to 
be  too  hi-h  or  too  low,  the  error  can  be  corrected  b>  loosen- 
int;  or  ti^;htenin^,'  the  nut  on  tiie  tie-r.xi. 

■  The  balance  proper  is  carried  by  a  sli.linu  hame  v  Inch  can 
1.C  ,noved  l„.ri/<.ntallv  into  any  position  alonj,-  the  bracket  by 
nuans  of  a  r.ick  and  pinion  actuated  by  a  sprocket-wheel  w.tli 
chdn  \l  one  end  the  fn.me  l,as  two  equal  arms  uith  a 
comn.on  Itori/ontal  axis  parallel  to  the  bracket,  and  cuh  arm 
has  a  stop  on  its  lower  surface  which  serves  to  hnut  the  oscd- 
lation  of  the  balance. 


il° 


I.\V\-U:i    ^f/',IR,ITUS. 


t'> 


ilk 


tfc 


t^'A 


■•*> 


1  lie  haiancc,  in  its  mean  position,  c(jnsists  of  a  main  trimi< 
with  liori/nntal  axis  ri;^itlly  coniKxtcil  with  a  xcrtical  sh)ttcd 
arm  anil  with  two  iniual  horizontal  arms  at  one  etui.  The 
conunon  a-j\  of  the  latter  is  horizontal  and  perpendicular  to 
the  axis  of  the  main  trunk.  The  hardem  d-su-jl  knil'e-ed-'-es 
ol  the  balanee  are  4  It.  centre  t  1  centre  and  n-t  in  hardened- 
steel  vees  inserted  in  the  ends  k\  the  slidin.;  frame  on  each 
side  of  tile   bracket.       Ihe  botlnn  of  each  vee  is  in  the  same 


Itori/ontal  line  icalled  the  a.\is  of  the  vees  i  at  rij^'lit  anodes  to 
the  bracket. 

A  bar  with  the  upper  poniMn  _i;radual( d  in  inches  and 
tenths  has  a  slot  in  the  lower  portion,  which  is  bent  into  a 
lircular  sft,'nicnt  of  9^  ins.  radius.  The  liar  slides  aion^'  the 
slot  in  tlie  vertical  -rm  of  the  b.d.mce.  ,\  r.ulial  block,  with 
tlie  holder  into  which  the  se\eral  \anes  are  screwed,  niovt-s 
alon^(  the  slot  in  the  circid.ir  s(.o,„t.„t_  .(,,,1  ,„;,^.  i,^.  clamped  in 
an\-  required  position,  the  anj^uiar  deviations  from  the  vertical 


1 


cohi/icii-sT  ('/  i\i/\u:r. 


i 


hciiii;  slinwn  I))-  L;railu,itiiin-^  nn  the  scL^iutiit.  'I'lu-  leiitic  of 
tin's  sc<,Miicnt  in  cvcr\-  ca-^c  C'lincidcs  w  itli  the  central  point  lA 
impact  on  a  vane,  is  in  the  \ertical  axis  o)  the  h.ilaiKe-arni, 
and  is  a'so  verticall\-  below  the  a\is  of  tlie  vees.  Ihn-  the 
jet  can  always  be  made  to  strike  the  vane  both  centralh'  and 
normally. 

The  scale-pan  Iuuil^^s  from  a  knife-e(l,L,U'  .it  one  end  o1  the 
lioriz  nitai  arms  of  tin  balance,  while  to  tlu'  other  end  is 
attached  a  fmt  iiointer,  which  indicates  the  an.L^ular  movement 
nf  the  balance  on  a  i^raduated  arc  fi>;ed  to  tin-  -lidin;^  tr.une. 
The  balance  is  in  its  niid-po-ition  when  Ine  ])ointer  is  opposite 
the  zero  mark. 

When  a  \ane  lias  beiii  secured  in  any  i^iven  position,  the 
prelimiiia''\'  atljiistment  <4  tin:  balance  is  effected  by  mo\in;.,f  ;i 
hea\\'  east-iron  disc  aloii;.;  a  liorizontal  screw  fixed  into  the 
main  trunk.  The  sensitiveness  of  the  balance  is  also  increased 
or  diminishetl  by  raisini;  or  lowerini,'  lieav\  wei'^dits  on  two 
\ertic.d  screws  in  the  tt)])  ot  the  tiimk. 

.\ssume  that  the  ,idjii-.tmeiit--  h.ue  all  been    nude  and  tliat 
the    jet,     l"v^.    2lo.    now    inipin;-;es 
norni.illv  upon  a  vane. 

lit    1/    Ik   the  weij^ht  rcejuired  in 
till-  scale-])an    to   briuL;    the   bal.iiice    ^^ 
l),ick  into  its  mid-position. 

Let  /■„  be  the  <ii//iti/  force  of  im- 
]).ict  determined  by  the  lialance. 

let  /,  be  the  f/nofiZ/t  ti/  (itrcv  a( 
iinp.iet  deduced  1)\'  the  onlinary 
1  irmul.e. 

/. 


.( 


/< 


I'lien    the   r.itii 


/•■, 


I'll;.  2IO. 

ma\-  be   called   the    coefficient   of 


impact. 

Let  r  be  tiie  \ertical  di-t.mce  of  the  centr.d  point  of  imp.ict 
below  the  horizontal  axis  of  the  orifice,  which  is  }(>  ins.  below 


'I 


m 


^te 


■y    ^::| 


;,7J  COhFI-iai.sr  OF  IMP-ICT. 

the  axi-^  of  tile  \'ccs.  The  ili^tancc  hctwcrii  tlii-^  axi-^  ami  the 
|)(inU  of  Niispcii'^i'Mi  (if  till-  srak-p.m  \-  24  ins. 

I.L'*.  ,■  he  the  \el<icity  witli  uhich  the-  water  issues  from  the 
orifice. 

Let  :■    be  tlie  \elocity  of  tlie  ji't  at  tlie  point  of  impact. 

Then 


/•;=^'>' 


.0 


0    liein-    the   'leliver>-   p'-r    srcotnl    and    0    tlie   anL;le    throui^h 
whieli  the  w.iti  r  1^  turned  on  tlu'  \ane. 

It  the  axis  ,  if  the  iet  at  the  point  of  imjjaet  makes  an  aiv.;lv' 
t^  w  itli  llie  hori/,,nt,d,  then 


Tlierefore 


r-'  cos  H  z=  V  —  i,  S  2^1^ //. 
ti 


I',  cos  (^  -z   2     C'^'  '^i"'" 
AL;ai'n.  taiNini;  moments  alxuit  /', 

/•'a   COS   ^(  36  4-11     --     //'.    24. 

I  li  nee 

' '  "  /•'/  '  ■    (^   .      . 

:r(h'  sin-     f^^.:    y) 

~  .     ii' 

wc.c  -All  30       yi  sm- 

A  heint:  the  sectional  area  of  tlie  orifice. 

A  lar^^e  nuniher  of  experiments  iiave  been  made  for  the 
purpose  of  determininj.j  the  v.ihie  of  <,  and  are  described  in  the 
Trans,  of  the  Royal  Soc.  of  Can.,  \'ol.  11.  iSQf),  and  of  the 
Can.  Soc.  of  Civil   Engineers,  Vol.  ,\1I       \'o  definite  law    o| 


RF..i<:rH)s 


373 


* 


variation  has  yet  been  frtum!.  but  the  followin^f  i;eneral   results 
ha\c  been  obt, lined  : 

'I'iie  actual  tbne  of  iin])act  i^  aluay^  nuah  le-s  than  that 
in. heated  b\-tht  irv'.  h"-ven  uiul<  i  tlienvi^t  tavor.ilile  londi- 
tions,  with  a  •'■yy  lar:_;e  coelhcient  i.f  \el(>cit>-.  the  theoretical 
force   of  impact  \v,is   foinul  to   exceed  the   actual  1)\-  3  <ir  4  per 

cent. 

The    ■  ■■■ftlcirnt  ^f  inip.ict,  <■ ,  .  increases  with  the  velocity  of 

the  jet. 

Tlie  coefticunt  rapidl>-  dinhnishe-  witli  the  an:,de  throuL,di 
which  the  stream  i-  dctlected.  It  1-  aUo  df  intcrr-^t  to  note 
that,  witli  small  an-les  of  d.tlection,  .,  was  L^reatest  \\  ith  a 
concave  parabolic  \ane,  le-s  with  m\  elliptic,  and  least  with  a 
circular,  but  that  this  order  was  reversed  when  the  deth  ctions 
were  lar^jer. 

8.   Reaction— Jet  Propeller.—  riu-    t. mi   Mactiim    is  cm- 

ployetl  t'l  denote  the  picssurt  upon  a  -urf.ice  due  to  the  ilirei- 
ti<in  and  velocit>-  with,  which  the  water  lc,i\es  the  surlace. 
Water,  for  rxaniple,  is-ue's  under  the 
he.ul  //  an<l  witli  the  velocit\-  :',  lat  con- 
tracted section  I  bom  ,in  orihi  e  of  section, d 
area  A  in  the  vertical  side  of  a  vessel, 
I•■ii,^  2  11. 

Let  A'  lie  tile  rciction  on  the  opposite 
verticd  side  of  the  vessel,  .md    let  Q  be  the   ([uantit}-  of  w.iter 
which  flows  throut;h  the  orifice  [)cr  second.       1  hen 

A'  _;  horizontal  chan.^e  .4' moun  ;ituni 

tlisrcpirdin!,'  the  contr.iction  and  puttini;  ,  1 

Tlius    the    reaction    is   tlmilile    the    corresp.)ndint;    pres<;ure 

wlien  the  orifice  is  dosed  {V.\.    1,  p    .>'>>i. 

A.L^'ain,  ht  the  vessel  be  j)ropeiled  in  the  opposite  direction 

with  a  velocitv  u  reiativelj-  to  the  cuth. 


Fig.   2ti. 


.5  74 


RI-.-iCriON. 


'I'hen    ,-'      -  a  is   tlit-   \Llni.it\'   cf  tlic    jet   at   the  cuiitracti'il 

1 

soction  r".lati\ol\-  to  the  earth  and 


A'  =  hciii/'Mital  chanije  nt  iin  niienluin 


A' 


,C^--, -"' 


The  u^et'u!  worl;  ihnie  h)-  the  jet 

=  A'//  =''c^/'(r',  -  !t].    .      . 
'I'hc  enerj;\-  earrieil  awav  1)_\    the  i-^->iiinL;  water 


(' 


I  leiKO 


the  total  eiler<jv  =  "  C^wi  r  ,  —   n)  +  "  (? 

K  K  2 


=       0 


and 


(2) 


(3) 


(4) 


(5) 


1*^ 


•''XXb' 


\^A 


tlie  efTl('ienc\'   =  '^ 


•     (6) 


Tims  the  mi  III  nearly  :•,  is  equal  to  ;i.  and  therefore  tlie 
larj^a-r  tlie  .irea  ./  of  the  oritue,  the  greater  is  the  efficieiicw 

If  the  vessel  is  dri\-en  in  the  same  dir^'etion  as  the  jet,  tluTi 
T'l  -|-  i(  is  the  relative  \eloeity  ot  the  jet  with  respect  to  the 
earth,  and  the  reaction  is 

A'        Iiori/ontal  i.h,in''e  i if  ninmentuin 


=  -  A:\\i-.  -■-  u\ 

ilisrf},%-iriliti.L;  the  eontr.ietion  and  putting'  i\.  —  I. 


(7) 


SCOTCH    71  RHIM--  ^'> 

g.  The  Jet   Reaction  Wheel     Scotch  Turbice  .-In  this 

torni  -r  iiH.t..r  the  v.  ater  cnlcr^  the  centre  ..f  the  whi'el. 
.])re;ul>  nut  r:i.h.ill\-  in  tuliular  passaL[cs,  and  i-'ie^  h-ni 
npenin- .  at  the  en.i->  tan-cnlially  to  t!ie  ihrection  of  rotation. 

Flu.  214. 


1 1...  212. 


"-tj 


Fie,.  213. 


Fie.,   21?. 


Fi"-.   2\2    npresont^   the    ■-iniplost  wli 


eei    o 


f  this   el, 


In 


Kntrlan 


1    it    i-    known 
.heel. 


liirke-   -    Mill,  and    m    ( .ernian\- 


Scf^ncr's  W.iter-wheel 

''a  reaction  ulieel  may  ii,iv>    -ever.il   tulml.ir  passaiH's  ,is  in 
I-io.  214.  while  the  ver;ie.d    ehaml.er  .VT  ni.ay  he   eylindrie.il, 

prism, itie.  or  eonie,il. 

The  Seoteh  o,-  W  hitelau's  turbine.  Fi^'.  21?.  <loes  not 
ciitfcr  cssenti.illy.  ex.cptin-  in  the  curved  ,irms.  from  the 
simple  reaction  whei  1. 

Let  /•  he  the  horizontal  dist.mce  between  ,iNis  of  orihce  and 
axis  of  rotation 
..   /.    .,     .     lK,»d  of  water  over  tiie  orifices  ulu!>  closed. 


hi 


h 


W  Is:* 


IS 

li 


3 7*^'  SCOTCH    TCRBISE. 

I.rt  ;  ■  he  tin- \ciocit\-  (if  tftlux    rclativily  t'.    the   tube  ulun 
Uu-  '  iiiticc-^  arc  nprii. 
"    tt    "      ■'    i.nrnj-~i)niii!ini^    iincar    xelncity   cit'    rotation    at 

t!ic  icntn;  ot  an  orifiLH.'. 
"   :■..  "     "    ab'ioliite  \  cli  icit_\-  of  ctfliix   .:-;    I' —  u. 
"    (.'  "     ••    (liscliar^c. 
••    A'  •  ■      •■    reaction. 
'I'lieii 

]'^  =  i;}(ir  +  2,i:k) (I) 

i.  heiiii;  the  i  oetlicieiit  of  dischar^^c. 
A  Is.,. 

'U'O     ,. 

(  /     —  u\    -  horizontal   niie.ir  chanije  ot  lllouieiltuni 
--  re.Kti"ii  [iri  nhicitiL;  rotation 

--   A' (2, 

'I  he  useful  work 

A-              '''^    /■ 
=    t\U    —  '  (      —   ll\H.     . Ml 

1  he  et'tk  iene_\- 

_    K/t    _  i  r  —  mil       2(  /'  -    It)// 

~  Ti'Qh""      Jfi      ""'V*'      ■    •    •    •    (-^^ 

Ajr.iin,  the  eflKiency 

.C//  I    "  ir    I  \ 

It-   (      /         .i,'//  I  / 

=      /•,''.lf4-'   .,— terms  containini;  liif^lier  ijowers  of     )  —  I   ■. 
.;,'■//  /    ■  \     '     //-  .-,      .^         1  ^^j  ^ 

Thus  the  efficicnes-  must  theoretically  increase  with  u.  but 
the  \ahie  of  //  is  limited  t)\- the  pr.ictic.i!  consjihration  th.it. 
even    at    niodcTately    high    speeils,    so    nuuh    of  the    lie, id    i- 


SCOTCH    TL'RlilSF. 


377 


absorbed   b\-   frictioiial    resistance   as  to   sensibly  diminisli   tlie 

efficiencN'. 

loteets  fif  tlie  reaction  uheel  are   tliat  its  sjn-ed 
l)!e   ami  that   it    admits   of  no   efficieiit   system   of 


he  serious  t 


is  nil  1st   unstable 


reL,nilation  for  a  var\-ini;  suppl}-  of  w.iter. 

\W  etiuation    ^4),  the  efficiency  is   a  maxi 
\aiue  of  ti,  when 


iTium,  tor  a    L;i\'eii 


J  •=  -  2 ; '//  + 


-«'  =  o, 


or 


J'=::r   mi    +    i    I    —  ,  Z^) 

l-^xperiment  also   indicates   that   the  best  effect  is   pri 


(5^ 


q 

;hen  the   linear  speed    of  rotation    (//){- 
lead  (//),  so  that 


that  due   to  the   t..tal 


an 


1  therefore 


r'  =  4'-,=c/'. 


>u 


bstitutiiiLj  these  \ahies  in  eijuation  'jl,  it  is  found  that 


2  I  2 


;vnd  hence,   in-  npiation  14"!,  the  'i.-.i.iniimn  efficiency  =  §. 

/>  ,  .... 

Thus,  one   third    of  the    he. id,    i.e.,   ^,  is    lost,    .ind    ol   this 

3 

.    TV     ( r —  //)■•••     //  .        .  ,         ,     . 

.imoimt  the  portion    "    =  =   -.  is  carried  a\v,i\   li\-  tlie 

efiluent   w.Ui'iin    its   eneri;)'  of  motion.      The  rem. under,  \-i/.. 

//         /,         2 

—  --=::      /'  i-,  lost  in  friction  il  resist, mce,  etc. 

^        9       9 

Ex.  A  reaction  wheel  urli  si.\  tubiil;ir  p.issasos,  each  of  4  scj.  ins. 
sectional  area,  passes  1 12.500  stations  o!  water  per  liour  and  makes  105 
revolutions  per  minute.     The  distance  between  the  axis  of  revolution 

.nil",  the  axis  of  an  ..ril^ce  is  z  In't.      I'l'ake  C;.  =  I.) 

/'  /L  _  '-     "  ^'^^     =  i  cu.  ft.  per  sec  per  orifice. 
I44~66i.60.60       6 


.?7'^ 


I M  PA  CI    U'HnFL. 


t  ■  i 


'^ 


Tlureforc  /'  =  30  It.  por  sec. 

,  2  .  ff .  2.105  , 

.VLj.ini.  ;/  :=  ,  =22  It.  per  sec. 

60  ' 

Hence,  if  /i  IS  the  lie.i<l  over  the  01  ilices, 

30'  =  22''  +  2.32  .  //. 

.iii'l  //  =  61  ft. 

:-:i  =  134',  Ib.s. 

=  6  X   13^',  X  22  =  171S5  ft.-lbs. 
=  3!.  H.F. 

i;iS}     _  II 

(>2\.i.bl  ~  \i 

10.  Impact  Wheel.  Borda  Turbine.  -A  jet  nioviiiL;  in 
the  diix^tidti  ( '(  Iml,'.  Jidi,  with  .1  \cliicit\'  ."■,  (  -  OL  \  mi- 
l)iiv_;c-  upon  .1  tlat  \ani',  (Iriviiii;-  it  in  the  direction  OE  witii  a 
vclocit)-  ;/  (=  OE).      Juin  CI:. 


-^■4 


The  rcictioii  r,ii  each  tube  =    ""  .\ixo 

^2     6  ^-' 

The  Useful  wutk 


The  ellh  icmv 


Fii-    2t6. 
I.ct  f'l  lie  the  ([iKuUity  Mt'iliiw  towards  ./,  .uul  »i^  its  ma>;«;. 

•'    0. ••  /;.      "     nr      " 

i  )isre,L,'ar(hni;  the  elTect  of  i^ravity,  which  is  equivalent  to 
tiic  ,i,-suni|)ti(in  that  tlie  ni'>\cnK'nt  of  the  water  on  tiie  wme  i- 
.scnsihlv  in  a  hori/ontal  plane,  and  aKo  (Hsri,'L;ai(hnL;  frictioti, 
tlie  water  h  a\-es  the  wine  at  .  /  ,ind  />'  with  .1  rel.iti\-i-  \elocit\- 
/'  -  yi,;,''  =  />j^  —  lE,  coineiiient  in  direction  with  ,  / /■'  pro- 
duced. 


IMI'.ICT   H'HhHL. 


379 


l)r;i\v  A  A'  and  /'/.''  parallel  and  equal  to  OM  — 
Complete  the  paralleto;_,n-ams 


.I/;,  1=  :■,■,  i-e])n;seiit- 


m  1 


r.i.-  and  A'-'.       Then 

lii-ccti^n  .111(1  inaL^nitiuie  tln'  at)->iilutt 


.■li)Lit\-  v.ith  which   the  water  leave-  the  vane  at 
A.  and 


/I//',  =  -•,".  reprc- 


;nts  in  direction  and  inai^Mn 


till'     till-  ali'-iilntt 


■loeitv-    with    which    the    w.iter    leave,   the    \ane 


at  /.'. 
Froin  the  trianL;le  ('(7; 


r^=  .-v+«» 


i//  cos  y 


■"ri  iin  1  '.e  trian 


de  A.r/i. 


r--\- 


2l 


U    -ill    <,> 


rniii  the  trian: 


de  /;/>'■// 


■-:=;    r-  +  /r-r^'' 


!(  Sin  (,'). 


Henc 


«(?-,  cos 


s  J'  —  ;<  —  / '  sin  0, 


Al-^ 


cos  <; 


w/.,  = 


T.'     Q: 


COS   f/)\ 

COS   H' 


Theretore  the  useful  work 


=:   W, 


-.2  _  -.  "2 


jt'  It 


cos  </»' 


T',  -  /^ ;  -ri)  ;C'(-',  COS  >'  -  ;/i  -t  (  C^  -  C',)^'  sin  0! 


cos 


—  V. 


=    U(i 


COS  y  ~  w  -;-  r  sin  (p) 


md 


^^ 


1 


hi^ 


380 

where 


IMPACT   liHlihL. 


Q  =  c?,  4-  a 


If  the  directions  nf  iiiMtinti  of  the  v.iiu- and  ( if  the  inijiintjinL 
jit  coincide, 

y  =  n  -\-  (,')  —  o     and      \'  -  :•,  —  u, 
.mA  thi  rctore  thi'  useful  ener^^y  imparted  to  the  \ane 


ri'  II 


^  ,-  „  '"•,  -  '<y-(Q  +  C^  -  Q,  sin  0). 

I'"or  a  maximum  effect  r',  =  3//. 

Sous  of  X'auis. — If  a  number  of  vanes  are  successivelv 
introduced  at  the  same  point  in  llic  patli  of  the  jet.  then 


Tat 


\^A 


S> 


;//,=    _(.>,     and     w,  =    ^CV 
Thus  the  useful  enerj,')-  becomes 

\ii.  C  .-,  cos  ;'  —  //;+( (.',  —  (J,]l'  sin  0; ; 

and  if  the  directions  of  motion   of  the  vane  and  tlie  impin^dn^' 
jet  coincide, 

;'=  «+0  =  O.    l'=  t\-  //. 

and  till-  iiseful  ener.L;y 

-  "."(•;  -"KQ+  (J,      0:Mn  01. 

i'lr  a  niaMiiiiini  eiicct  f,  r=  2/1. 

/■Vi'W  III  (hit  riny/ioii. — If  the  whole  of  tlie  water  flows 
away  in  tiie  direction  (>./  so  tliat  C',  =  O  and  (,',  =  Q,  the 
useful  encr^)   for  a  siiii^^lr  v.me 


«■(?  u 


eos    Hi 


'  I 


-III    I, 'i 


and  the  useful  cucrtj)-  for  a  siius  nf  \ancs 

r^'Cj  ....       , 

=         u[-,\  cos  y  —  u  -\-  I    sni  <;>, . 

For  a  j^ivcn  value  of  (p  this  last  is  t;r<jattst  wlun  y  { --  ^^  -f  '•'') 
—  o,  ami  then  fore   f  =  r,  —  ;^      Then 


maNimuiu  useful  encrj^^y 


^■(J 


ii{:  rt  {  I  -j-  sin  0), 


which     increases     with    (,'>    "r    as     th<;    an^le    of    exit    ('/LJ 
(^—90    —  <;>;   (iiininishes.  iiulicatinL;  that  it  is  ailvantaycous  to 
curve  the  outlet  lip  of  the  \ane. 

Denote  the  exit  an^le  hy  <■,  I'i:^'.  2;y.      Then 


anil 


p  =  ?/'  -f-T',-  -  2it: ,  cos  )' 


:•,'  =  u'  -     I''  —  2«J'cos  (•. 


.hus    the   useful   ener/v   imparted 
to  tile  vane 


/^i . 


cos    )'   —  //  -|-    I'  COS  f). 


If  <  is  so  small  that  cos  ,■  —   l ,  ap- 
prnximately,  then  the  useful  enerj;y 

=         «(t',  cos  y  —  It  -i-  I  t 

—         «(:•,  co-- 

-t    »«»-+-«',"  —  2«7',  COS  )',K 

'Ihis  is  };rcatest  and  —         -'    when  //    -  ir,  sec   y.  w  Inch 

is  the  best  speed  of  the  wheel.      In  such  case  the  whole  of  the 
jet's  eiur-f^y  is  transformed  into  useful  work. 


i\  J:J 


5S2 


Bi>h:/).-is  /iKRiNr. 


Ill    the   Miiiplcst   kind   of  inii);n.t   wIk-cI   the  jet   strike-   tlie 
\.inc  lUdii-  or  hs>  piriKiuiicularly  and  spreads  over  the  surface 
in  all  directions.      Wheels  of  5  ft.  diameter  are  used 
for  falls  ,,f  from   10  to  20  ft.     'I'lie  vanes  are  r  5  ins. 
X    ."<    ins.    to    10   ins.    measured    radiall)-,    and    are 
inclined   at  from  50    to  70    to  the  horizon.      'I'he 
water   -trikes   the   vane   in   a    direction    niakin;^'   an 
an-lc   of  fii.m    10    to   2u     with   tlie   hori/on,    i.i.. 
nearl\-  at  rii;ht  anL;les. 
In    a    i^orda   turbine     l'ij,r.  ji<j;  revolviu^;   about   a  vertical 
axis  (H\  tlie  \ane-   are   curved  and  the  wat  t.  as  it  tlows  over 
tlieni.  act-  ].riiK  ijiall)  b\-  i)re--ure.       I  he  v;     es  are  set  between 
two  concentric   drums  uhicii   should  be  of  considerable  ileinh 


Fig.  21  S. 


ti^:i 


'  "■■  -'''■  Kic.  220. 

and  should  ha\.  a  ....-e  mean  diameter.  Hor.la  found  that 
with  sucii  a  turbine  an  cfticiency  of  ;;  per  cent  could  be 
obtained  under  favorable  conditions.  As  tjie  water  passes 
thr  .u^;)!  the  turbine  the  lluid  particles  move  wholly  in  cylin- 
drical sm-fac.-s,  aiu!  there  i-,  little  if  any  chan-e  in  tlie  distance 
of  a  particle  from  the  avis,  Thu-  the  effect  -f  c.  titrifu-al 
force  may  be  disre;,Mrded. 

Let  \-v^.  ;:20  l)e  a   section   of  the  vane  at  a  distance   from 
the  a.vis  et|n.il  to  the  mean  radius. 

I'lic   water  strike-  the  \..ne  at,/  in   ilu    dio.ti,,,,  .  ju 


/(()AV',-/'s    Ti.  K/IIMF. 


i^3 


uiKin  tlu'  \.mo  tliiuu;;li  a  M-rtic.il  ilist.mcc  //,.  .itui  is  (iiscli.irL,fcil 
;it  /  ill  tile  dircctiiiii/'/'  w  itli  a  vcl'n.ity  ;  .. 

Let   ;'|,    ]\\)L    .iif   ivlatnc  vclocitiis   ,/,/  at  <^  and  /i,^  at   /, 
ri  spcctivcl}-.       I'licii 

If,  as,'aiii,  till-  allele  ..fcxitr  at  /is  •<(>  small  that  ens  f  —   i, 
a|)|ir<i\itnatcl\ , 

'••.•        '  \        "• 

Suppose   tliat  tJK-  watiT   Icaw^  the  turliiiu.-  witlnnit  ctKrtjv. 
i.e.,  so  that  i\,  =  o  ---  /'.,  —  //,  thin 

=  11-  4-  '■•,'  -  -""'i  t^""-^  r  +  -.ir^'. 


and 


2tl7\  cos   )' 


r.    2.C(//-|-//,) 


or 


iiv,  cos  f       gH, , 

an  equation  f^ivinj,'  tiic  l)cst  s|)fcd  of  the  turbine. 

//  i-  tlie  head  leijiiired  to  ^-ive  tlie  \eiocity  .•,  at  entrance. 

//,  is  tlic  total  head  inuler  which  tlie  turbine  works. 

There  should  be  n<i  loss  in  shock  at  entrance,  and  to  insure 
this  ,t,f  (  --  /■,',  the  relative  velocit>-,  must  be  t.in^'ential  to  the 
lip  at  (/. 

I  he  111  I  an"  11'   .i    i  .  !  Iumi    ■  -  i  w h   Ii\- 


^ 


sin  >) 


<>t  It  sin  ;-, 


i.S  "S 


3«4 


BORU.rs    TLRBINE. 


lilt  a  —        coscc  ;'  —  cot  y. 

Since  /t  —   /', ,  the  tri,mL;lc  />//  is  isdsceles.  atiil 

<* 
'.:,  =  2//.  sin     . 

■  2 

'ilic  useful  '.vork  =  "     '/  f  —  —  ), 

;;  IjcIiil;  the  efficiencx'. 

Let  A'  he  tlie  mean  r.ulius. 
"   /    ••      ■•    water  thickness,  nieasureil  railialK', 


lien, 


J  T     .  <   ,    in  I—  (K 


Allowance  ma\-  be  made  for  tlie  p-'-incipal   Indraulic  resist- 
ances (friction,  etc.)  bv  takins' 


/   '  '    t'>  reure'-ent  the  loss  of  head  u|>  t<i  the  inlet,  and 
-  2  2_i; 


in  the  wheel -passages. 


'Jt 


t:A 


"*, 


and 


.'  I 


(•+/.)2;^-/^ 


21**         2l'' 


/,  and/,  beinf;  coefficients  to  be  determined  by  experiment. 

I'siialh'  /!,  \aries  from  .025  to  .2  and  upwards,  an  average 
value  beinj;  .125,  anil/,  varies  from  .  1  to  .2. 

The  normal  distance.  V'v^.  221,  between  two  consecutive 
vanes  should   be     •  the  stream's  norni.d  thickness  between  llie 


m  RDis-s  i-ynt-hLs. 


38s 


\.incs  i.e.,     >   ',1  X  the  normal  thi(.kii<-~s  of  the  strc.im  bt-fcn 
iiui)act. 


Fii;.  22r. 


V\r,    222 


liunlin's  (l-i^'.  222)  is  anion-  the  \,v^i  d"  iinpait  wliccls. 
ililVcriiii,'  only  troni  the  simple  Horda  m  r<icivmL:  tlic  'Aater  at 
several  points  siiiuiltaneously  ami  in  ilistribntiii^;  the  riiitlct 
opening,"-    n  three  coiuentrK   lirrles. 

Ex.  A  5-11. r  Hoida  tiirbim-,  .  !  4  U  mfin  diameter  and  4  ft.  depth, 
works  under  a  total  head  it  -o  ft.  1  In'  diiei  tion  of  tlie  jet  before  inipaet 
is  inchned  at  33"  33'  {y\  to  tlie  hoi  izon,  and  the  angle  of  exii  (<)  l^  h/S'. 
The  jet  delivers  3  cii.  ft.  oi  water  per  ser.md.  Find  (,;)  the  1h  st  speed 
of  the  turbine;  (/i)  the  lip  an^lc  ,t;  (, )  the  velodty  :■,  .  .  f  the  water  as  it 
leaves  the  liirhiiie;  (,/ithe  hvdr.iuhc  eiriLienrv;  <n  the  pr.xtu.d  etbl- 
cieiicy. 

(<i)  >o  —  4    -  I*""   =  head  lecjiiired  i<>  produi  e  j'l  -  . 

'I'liprefore  v,  =  3:!  fi.  per  second. 

The  best  speed  i>  then  j^iven  hy 

,„  ;/  .  -,2  .  ros  33°  33'  =  32. 20. 

or  w  .  32  -    p. 20.        and     u  —  24  ft.  per  second. 

60  X  24 

The  nnnd)er  o(  revolutions  p»  r  minute  =   , ,   ^       =  •Mi'f 


id) 


siii((i  4-  y)  _  « 


Bill  it 


Vi       32      4 


=  cos  y  4.  rot  i«  sin  y. 


a  kI 


(ol  (180   -  <«)  =  <ol  33"  33'  -  ',  coscc  33'  33'       .1509. 

4 

■'   -  98*  JS'- 


Fi 


!86 


DAS'.-t/DES. 


mm 


(t  I   Assumint;   I',  .=  u,  then 


■X 


^ti^-A 


'■'  "   -"  "'"  ',  ^  4S  sui  ./  34'        i8  X   '  =  S  ft.  per  second. 


■95- 


!./>   'I  ii  -  hy!ir;iuli(   etiiciency  -  i  —    F-^  _  L? 

64.20       20 
(<•)    If  '/  is  the  [iraetical  ciricieiicy, 

aiul  n  -  .772. 
naHauh;.~Jh^.^  arc  uliccls  capable  ,.("  rcvolvini;  at.,uit  a 
vertical  axis  and  consist  nf  tu-..  casjn-s  wliich  arc'  nmrc  „r 
less  in  the-  f..rni  of  inverted  truncated  cones  i  I-j-.  o,,^  .„„, 
which  enclose  a  space  divr'.^d  into  a  numl.er  of  w:iU:r-i)assa-es 
hy  vanes  uhi.h  nu>-  I.e  Hat,  spiral,  ..r  screw-shaped.  In  Uie 
wheel,   described    by    Helidor   the    inner   ca^in-  u  ith  the  vanes 


y\i.'.  224. 

attached  is  nude  to  cK.seiy  ht  the  ,uuer  conical  casin^^  which 
1^  hxed.  In  .mother  (orn,  .,t  1  ..maule  the  vessel  is  divided 
.nto  two  e.jual  p;ut.  by  a  verta.tl  partition.  Thus  in  wheels 
ot  tins  type  the  u  tter  approaches  the  axis  in  its  descent 
developinfT  ^  eentrifu-.il  i,„ce  uhich  must  ,c  taken  int.. 
account. 

Consi.ler  the  case  ol   a  Dana.de  with  double  cnical  casin- 
and  lilt  vertical  vanes.  Fijj.  224. 


» 


~i 


DANAintS. 


3«7 


The  relati\c  vclncitirs  /'  /'  arc  f\icicntl\-  at  rit/lit  aiu'lcs 
t.d  till-  cDiTL-spuiiilinL;  iKrii)luTal  velocities  at  inlet  and  exit. 
Theretorc 


/•,-'  + 


//,'      aUe 


.\!sn.   it  //,  is  till-  depth  cifthe  uhei-1. 


17     r 


^A"- 


=    7.'    +/'.- 


/<,'  —  «.. 


",."  —  ", 


the  term 


due  til  the  elTect  <if  centritui;.i!  fo 


I  IciKi 


//,-  —  //., 


Hid  tlu'   DieehallK  al  ellei  t 


-,  a 


-L-  //.  - 


«'C? 


/'/^/'-.-e/T,/.—  Ihis  t; 


rni  I  it  inipaet  w  heel.  I'  i 


(// -  —  u.;\. 


consists 


nt  a   niinilKT   o 


t    tlnats    t"i\ed  to  .1  \  ertK  al 
citllcr  httcd  into  a  w  ell,  a  small  c.  leatanee 


^hatt.       11 


le  wlut'l  IS 


leinLT   ailoueii,    or  it  is    iji\en    a    larijcr 


diameter  ajmI  is  plaeeil  iu--t  below  the 
uell.  The  \^  atiT  is  l)roUL;ht  ahav^f  a 
l)ro[)erl\-  ilcsij^ned    race,   enters   the   well 


tani^entialh-   with    cnnsiderabl 


e    \eloeit\" 


■Ji 


Fio.  225. 


and   aiijuires   a   rotai  y  niotion.       i'lnis  it 

.lets  upon  tile  floats   both  b\-  inipai  t  A\\i\ 

by  pressure.     The  efTicienev  •  it  the  v.  heel 

is  small,  as  a    hirtje  portion  of  the  watiT 

esc.ij)es  without  pi.idiuiiiL;  its  full  elVect.       I'raetieal  experience 

iiuHcates  that  tlie  i)est  speed  ottl'e  middle  of  tlu'  tloats  is  about 

"//,■  ////;■// of  the  velocity  of  tlic  current,  and  that  the  efTiciency 

varies  from   i  5  t«>  40  per  cent,  but  r.trely  exceeds  30  per  cent. 


k 


1  1 


i     ye  I 


388 


iMP-icr  ()\  ':ch;rfii  r  i\7:s. 


m 


■■Bfe 


It.  t^^ 


II.  Jet  impinging    upon    a  Curved    Vane   and    deviated 
wholly  in  one  Direction— Best  Form  of  Vane.  — I. n  the  in, 

of  sectional    ana   ./.   nunino    ;,,    [l,^. 


(iircctiiiii    .l/<'    with 


a    \ch)Cit\-    T'  . 


r\       '  dv'wc  the  \anc  .!/>   in    tUc    (iii\-cticiii 

'    f:       «        ....  , 

^U    with  a  \cliicit_\-  ;/,    1-il;.   22h. 

UN    I  1  •'!<'-'  -'''''  t-'i  rcjircsciit  r^  in  dircc- 

/■     tiuii  an.l  niaL,niitiule. 

1  .ikc  .  /( '  t"  rc))rost;nt  u  in  dim  - 
tidii  .m  1  in;;tj;nitudc. 
Join  c7.'. 

I  lu'ii     (  /.'    c\-idint!y    rc[)rL- ,rnts 
/ ',  the  \  cloiit}- ottlic  water  rchitivcl)- 
^  to  tlic  \anc,  in  direction  and  mai^ni- 

■■■"•■  22'''.  tiidc.       \(('/:\<.   p.irallel  to  the    tan- 

S,'ont  to  the  vane  at  ./.  there  will  ')e  no  ^udden  elian;4e  in  the 
dircetioii  of  the  water  as  it  ^^tnke-  tlie  vane,  and,  di-re-ardin^u: 
friction,  tlie  water  will  tiow  ,doii--  the  \ane  lidin  ,/  to  /)  uith- 
niit  any  chanL;e  in  the  iiia-niti  le  c.f  the  rel,iti\f  \el.icit\- 
/-'(  -  C  ]■>).  '1  he  \ane  is  then  -ai.l  to  •■  reeei\e  the  water  with- 
out slidck.  " 

Atifain.   from  the  tnaiiirle  ./AT,  di-notmt;    the  aiiL;le>  />'./(', 
AKC,   .■/('/>'.   In-..',    /.',   t",  respectiveh'. 


u  _  y/( '        sin  /.'  sin  /)' 

'•,  ~  AB  ~  sin  C  "  ^xniAT-flf)' 


(0 


and  then-fore 


cot  />'  —     'co.soc  A  —  cot  A,    .      .  ('>'\ 

u  •      •     \-) 

a  formula  f^nvinj,'  the  angle,  between  the  lip  .uicl  the  dine  tion 
of  the  inipinerinf,'  jet,  which  will  insure  the  water  heiiiL;  received 
"  w  ithout  shock.  " 

III    the  direction   of    the    tanj^'ent   t-   the   \ane    at   t\    t.ike 
DE  ^  CB{=^  V). 


IMPACT  ON   cLRynn   y.^NES. 


3«9 


Dr:i\v  /V'"  ]xirallcl  ami  o(iu.il  to  AC  i=  ii). 

Cninplctc  tin-  p,iiMllclii.L;r,im  /;/'. 

'I'licn  the  (li.iLi'iiial  Pi i  c\itlrntl\  rci)riscnts  in  dircctioi) 
and  inaL^nitiidc  tlu'  ah^olutu  \clucit\-  ;■,  with  wliicli  the  water 
leaver  the  xanr. 

]  )ra\\   ./A'f(|ual  and  iiarallcl  tn  Pd  i  =  r'_,). 

|(iui  !>/\.  Then  /.'A'  rc[)rc^(.iUs  the  total  .liaiiL^e  of  velocity 
between  A  anil  /'  in  direction  and  nuiL^nitude. 

Thus   if   A'    i--    the    resultant    pressure    on    the    vane,    then 

A'  =  VI  .  r,K. 

Let  Ml.  lie  the  iirojeetion  of  /.'A' upon  .IC. 
Then    Ml.    repre-.ent<    the    total    JiaHLje    of  M-locit\-   in    tile 
ilire.  tion  of  the  \  ant''s  nio;ion. 

l.it   /'he  the  pressure  upon  the  \ane  in  thi>  dii-cetion. 
rnen 


y  -^  ;;;  .  I..U. 


The  iisifiil  zvork  —  Pii  ^-  iuk     f  U  ~  "i 


'  1  '  i 


(3) 
(4) 


The  total  ,i:ail(tblf  :oork  =  -  A 


•  1 


mu  .  LM  f,'  -  ''Z 

g        2 


(5) 

(6) 


A    ain,  join  (.  A  . 

Then,  since  AC  is  e(]ual  and  parallel  to  Pl\  and  AK  to 
/>(,,  the  line  ^  A'  i-~  equal  ami  parallel  to  /'/•..  and  i>  therctoie 
eijual  to  Cr<. 

'l"luis  in  the  isosceles  trian^;le  CPI\.  I  I!  h  equal  arid 
parallel  to  the  relative  velocity  /"at./,  (A'  is  ecjiial  and 
parallel  to  the  relative  velocity  /'at  /',  aiui  the  t)a->e  /i/\  repn  - 
sents  the  total  chanL;e  of  nvition. 


I'     ■< 


J    -V 


.^90 


iMTMcr  o.\'  CA  Kyi:i)  r^NHS. 


I.it  .•)  Ik-  the  an-lc  thrMU-h  \vhidi  tlu-  dircLtiM,,  ,.f  the  watcr 
is  .Icvi.itcd,   i.e.,   the  ,m-k-  iKtuecii  .•//,'  ,,11. 1  ./A'         lUvn 

/'•-  =  t'A'-  =  ^M"   ,-  ./(■••  -  2.//V-  .  .-ic  ,„s  ,.i  -f-  ^) 
ami  also 

=  '','  +  'r'-  J,-y.  C(,s  J -.       .       .       (8> 

1  li'IK'c 

--  '  =  «:r',  CDS  .•;  _  :■„  cos  (A  4-  rf)'.    .      .      ,,_;,, 

If /'//is  (Iraui)  parallel  to  the  tan;;cnt  U  />.  /.'A' evident;> 
bisects  the  allele  between  AT  and  /!//.  and  tlu,  ui'^lc  is  e,|ua'l 
to  the  anL,de  between  the  tan,i;ent>  t  1  the  \an-j  A^  A  and  /). 

I.et  a  be  the  an-le  between  the  normals  at  A  and  1). 
'riini  the  ant;le  A'c '/>'  ^  a,  and 


the  an-le  C 'A'A'  =  ^(  180'  -  ,r)  =  90°  - 
riiiTefore 


a\ 


I'K  -=  2r/.'(^cos  90°  -  J  ,,   . ;  -  sin 


I  lenci 


■.^* 


k.:} 


A'  =  HI  .  JIK  =  2»i  /  ■  sin  "^ . 


(10^ 


I.ot  .\-.    )-l„-  the   cnn,i„,nents  ,,f  A'  m  the   direction    of  the 
normal  at  A  and  at  n-lu  an-les  t.-  this  .Invction.        Ihen 


-^  =  A'  cos       =  ml'  sin  n. 


}    -A   sm   ^     „   2»,l     -iir  ^        .;;../,,    -COS0I 


(12) 


IMI'.-ICT   0\   c;  ATf./'   yANES. 


391 


Tlic  cffRicncv  is  a  niaNiniuni 


.hen 


^1 


(t{  1  II  I 


(13) 


Tlic  cffR-icncv  is  nil  when 


Pu  =  o,      i.e.,  when 


;^   =  O    (ir 


P  =n  O. 


V'4) 


Til   the   latter   case,  since   /' —  m    I.M,    the   projeition    I.M 
-I  he  ml,  and  therefnre  j.,,^   ^^t. 


UK    niu-t     I) 


It     rii'ht   k; 


.UlLMe- 


t'l  Ai ',  a--   in   I-'is. 


\    x^ 


'rheani-le  ,-ic7»'is  n( 


=  1X0" 


Therefore 


ti    _  sin 

7',  "  sin 


sni 


ACJi 


\2 


a 


sm 


If   /.'A'   IS    ], 


lUel    to 


AC,    Iml;.    ^2S,    then    the 
angle 


I 


i 


ACP  =   -fi8o''  -  <'0  + 


=  90°  + 


a 


and  therefore 


u       sin 

V.  "  sin 


,/r/.' 


cos  I- 


("+'^) 


COS  tr 


392 


IMI'.ALl    ON   iAMl'l-D   yANES. 


l.ct  tin-  direction  of  the  impint^ini^  jot  be  tan^'ential  to  tlie 
\.inc  at  ./,  l"i^'.  2zC),  ami  let  the  jet  and  vane  move  in  the  same 
direction.       riicn 

(    =  .',  —  N.      w  -  —  .-/i:',  —  //); 
P=  y  =   '   A{i\  —  u^'i  I  —  cos  tr)  —  2'    A(:',  —  ;i'('sin-     ; 


||| 


usfful  work        /*N  —  2  — ^J;/(:',  —  //V  sin'^  -; 

,-^   .                    .''( r',  -  «)•-    .   .,  ^i- 
etTicieiic\-    -  4  -  .j snr     . 

This  IS  a  niaxinuini  ami  etiual  to         sin-       uiien  :■,  =r  7^1/. 

-/ 

'1  hc->e   results   are   idcTitical    with   tho-;e  for  a   coi  cave   cup 
wliL-n.  ,1  --   I  .So'. 

Instead   of  one   \atir    let  a  series   of  v.'.iu's   be   successively 
introcluceu  .li   short    nutr\al-.   at  the  same   5  in  tlv  nath  o) 

the  jet.      Then 


t:':i 


'^, 


and   lienci.-  the   pre^-^ure  /'.  usei'ul  work,  .ukI   eilicienc}'  respec- 
tively become 

A7;  .  LM,     -Ar,'      — -;        '      .,    '. 


N.15.       I'Victional  resistance   may  be  taken    into  account  b\- 

assumini;  th.it  it  absorbs  a  fractional  ])ortion  ot  the  head  coire- 

sponiliTii;  to  the  velocity  of  the  jet  relatively  to  the  surface  over 

J- J 

uhicii   it   spreads.       I  lui>   the    loss    nt    head  =  /        ,    .iiul   the 

-    2j; 

.1'' 
corresponding  loss  of  enertiy  — -  U'O  f 


tiXAMPin. 


3» 


I  \ 


Ex.   A  Curved  vane  in  the  lorni  ol   tlu-  quadrant  of  .t  rircK'  rrrcucs 
Tvilhout  s/iock,  at  an  ed^e,  a  stream  oi  water  lluwinj,'  .it  the  rate  of  u  it. 
per   second,   which    drives  the    vane    with 
a  velocity  of  4   f;.    per  second    111   a  (hrec- 
tion   ntaliin};  an    .iri};le  of  60°  with   tlie   le- 
ceiviM;;  edye. 

At  the  receivinf;  edp;e  tiie  triai:>;le  ahc  is 
.1  triangle  of  velocities  in  wiiich  she  angle 
ubc  —  \lo\ac=  12  ft.,<j(i  —  4  r'.,an(i  A  ■— r, 
tfie  relative  velocity  at  a  which  must  be 
parallel  to  tlie  tan)fciit  at  a,  as  there  is  to 
be  no  loss  in  sliork.     Then 


128 


4'  +  /-■■■'  - 


;.4  I    cos  120  , 


V  ^  9.48<>i  ft.  per  sec(jnri. 

Also,  if  y  is  the  an,i;le  between  iic  and 
the  receiving  edge,  then  the  angle  ca/i  =  fvj"  —  y,    and 


9.4891  _  /'i         sin  (60°  —  I'l 


sin  60°  cot  y  —  cos  60° 


4  ii/)  sin  y 

or  rot  y  -  ],.}\<<(i     and  ;•  —  16°  47'. 

.\t  the  discharging  ed-e /!,•/;/•  is  the  parallelogram  of  velocities  in 
which /^t^  parallel  to  c/',  =  4  It., //•,  taiigental  at  /,=  9.4891  ft.,  tlie 
relative  velocity,  and///  i>  the  absolute  vclocitv  in  direction  and  magni- 
tude with  which  tlie  water  leaves  the  vane.  I.rt  the  angle  /i//:  =  ^, 
Then 

Ti'  -  4"  +  19.48911'  —  2  X  4  A  9.4891  cos  30°  ^  40.299J, 
and  t'j  =  ^1.3481  ft.  per  sec. 


.■\gain. 


9.4891       /i      sin  (<?  +  30  )                                    „          , 

-^  —  =  ■;  ,  =           .       ■'  =  cos  30"  +  sin  30°  cot  i5, 

4           //i-             sin  (»  J                 J               ' 

cot  ^  =  3.0126.  or     ^  =  18°  22'. 


^ 


)••: 


12.  Tangential  or  Centrifugal  Turbines. — Suppo-^c  tliat 
the  vane  .//'  i^  cntistr.iiiuil  t"  icnoKc  .ihnut  a  \crtical  axis  () 
witli  . I  constant  .int^uln-  clocit}-  <.).  It"  (V^,  (^C'arc  consecu- 
tive radii  and  if  /W  is    '.awn  at   ii'ht  aiu'lcs  to  0(>,  then  the 


.;94 


r.lNGllNTMI.    TL  h'tUSFS. 


work  cf  tlic  cciitiifii^Ml  force  as  a   mass  /n  ni  ilmd  muvcs  rroiii 
/'  to  tlic  consccii'ivo  position  Q 


=  unJ~r  .  OX 
—  viwrr  .  ,i)\ 


uhcfj  ( V  =  r. 


--}T> 


*  mar 


Fli;.  230. 
The  total  work  in  tlu'  niovcniont  from  .-/  to  /) 


:^      /       »iui-r  lir 


tfiar 


('-.'  -  >-;') 


M 


U      u    li.iiii;  the  linear  voU. cities  at  ,/  .in<l  P  respectively. 

ir  tlu-  llou  is  trom  .■/  to  /'.     ■'  ~     'is  evidently  a  ^,Min  of 

lieati,  uiiile  it  is  a  loss  of  heail  if  the  flow  is  from  /)  to  yl. 

In  t  inj,'ential  or  centrifii^^al  flow  turbines  a  number  of  vanes 
are  encased  and  have  concentric  inlet  an«l  outlet  surface-^. 
Ihe  (low.  which  is  more  or  less  radial,  is  AT.vf/vA  the  axis  in 
the  inward-flow  and //<'///  the  axis  in  the  outward-flow  turbiiu  . 
Since  the  axis  of  rotation  is  vertical,  the  effect  of  gravity  may 

be  disiei'.tlilf i| 


TANGhNTl.4L    TLRttlKF.S. 


.>95 


If 


tivcK 


are   the   raiiial    comjniiient'^   of  r-,  and    :\^  respei 


vj  =  I',  sin  V     and 


/ ',  sin  fi. 


I: 
Fig.  2-<i\. 


Fir..  J12. 


Then,   by   the  condition    of  continiiit\    <jf  flow,   and    dis- 
re^arihn^  tlie  tliiiknc.ss  of  the  vanes, 

2ni\ii^i\    —  "Tr,rf,r',  sin  y  —  Q  —  2nr,i/.,7\    =  2nrjijV^  sin  fi. 


and 


</,.•■,  sin 


Q 

271 


/.,</./ '.,  sin  fi. 


(0 


$ 


(/,  and  (/,  heinL;  the  inlet  and  ontKt  deptlis  of  the  uhccl. 
First.    Disregard  Itydraulic  resistances.      Then 

=  x\'  -  2//,f,  COS  y  -f  «/, 


(2) 


or 


,;  •}   2//,f,  cos  ;/. 


(3) 


m 


396  T/INCjHNTI.-a    TURBINES. 

Also,  (Vfim  tlic  ir'\di\v^\c /l-/t, 

•  V  —   I './  +  //.,-'  —  2  /  '.,u,  cos  /i. 


1  IciK'c 

///i'  useful  zvork 


(4) 


-  "^  (".• 'l  '^O'^  .»'  +  ";;  '  \  COS  (i  -   7//) .  (5  j 

The  encr},fy  carried  away  by  tlie  water  on  leaving'  the  tur- 
bine shoii',!,  of  course,  be  a^  small  as  possible.  Two  assump- 
tions are  u-ually  made  in  practice,  viz., 

citlur  u,  =  [', , 

and  then  also,  by  eq.  O;.  u^  =  J",,  sn  that  tiie  triant,'les  fUi 
.md  </((/ are  isosceles; 

so  that  the  flow  at  outlet  is  radial,  or  ;■„  =  tv".  and  therefr,,-e 
tile  tant;ential  component  of  r.,,  or.  as  it  is  called,  the  outlet 
velocit)'  of  whirl,  is  nil. 

././c/'//;/^'-  (/it    ,issui»/>fii>//  tt.,=  l\,  in   which   case  tlie   tri- 
an},'les  /X//  and  (/o/are  isosceles,  then 


••'1  =   '',  -   y  ^cc  ,'  =   '^H.,  =  '.'  l\. 


(6) 


Hence  ec].  (,1")  becomes 

<_' 

or  ^,v/,  sin  2y  -  rl^it,  sin  fi, 

and,  by  eqs.  (5)  and  (6), 


^"^  '    •    ■  ^1        2  CO.S  y 


///,• 


(7) 


uuluf  :.;.,i-    _  "  ^[^'l  _|_  „^J  COS  /:<  -  uA 


^     2  ^         r*C(xi'y^ 


(8) 


T.^KCF.NTt/tl.    TLRBINES. 
The  corre^r       "'"K  tftKifiKy 


r.r 


sin- 


39? 


(9) 


r--  COS'  y 

Aiit'f'tin^!;  tlh-  iissmiipthiit  '5  —  90°.  then 

7V'  cut   /}■  —   .-■„  Lol  /i         /^,  / '_.  Cd-^  fi.      .       .       i  lOi 

Kq.  (l)  now  heomes 

Q  r- 

r  (/,:■,  sin  y  —  -      =  ':,,/,//..  t.ui  fi  -^  -  -</.m,  tan  /i,      ( I  I ) 

and,  b\-  ciis.   (ij,  15;.  and  1  lO). 

«'<j  ii'() :•,-'/-, V,  sin  Jr 

///.•  useful  work  =.     ^^  ,/,7'   cos  ,.  =     ^^    ^   —  ^^  ^^^^^  ^.    .      (12) 

The  rorrespdtidint^  cfficienc}- 

=  '-''''>  ^'"-•^'  (>,) 

r.^ii.,  tan  (i 

Sciond.     l  he   principal   liydraulic  resistances  may  he  taken 
into   account   by   takini      he    loss   ot"  iiead  up  to  inlet  —  f,^ ^, 

and  the  loss  of  hea<l  in  the  \\  heel-passajjes  —f^    ^-,  so  that  tlir 


U 


tot.d  loss  due  \n  the  resistances  in  (juestion 


Eq.  (2)  now  becomes 


(141 


CS) 


an 


d  if  //  i-  till    lu  ad  o\fr  the  inlet, 


.     .     (i6> 


•1 


39.S 


hXAMPLES. 


N 


Kx.  I.  A  (entrifugal  itnv.ird  (low  turbine,  with  equal  inlet  aiiti  ,,i,tl<-l 
depths  and  workinjc  uiuler  tiie  head  of  -oo  ft.,  jjasses  i  cu.  ft.  of  wMter 
per^second.  The  .u.k!.-  ,.-  is  15- ;  4r,  =  5^., ;  and  it  is  a.ssiM;ed  that 
«j  =  /',.  Find  I.J)  the  |n-tipheral  speed;  (h)  the  lip  anjjle  at  outlet;  (< ) 
the  energy  carried  away  by  the  water;  (,/)  the  ener^v  lost  in  hydrauhc 
resistance;  (£•)  the  useful  work  :(/)  the  elficiency.  (iJisre-ard  thethnk- 
ness  of  the  vanes. ) 

sui  15°  .-  .359.     ,os  15'  -  .966,     ,ind  lit    /;  =/,  =  .125. 

'   "." 
'   ■''  8^,4   "=  -°°       Therefore  -•,  =  io6)i  ft.  per  sec. 


(<») 


By  eq.  114). 


/','  +  K. 


II,-  -    ^i';- 


.'•■•■ 


.It,   Cos 


'*■-?''■•=;(:)'"■•= 


o8«,« 


so  that 

A  Is.., 


Therefore     cot  (180 


^1  -  I.972W,  =  106J, 

«]  =  54-09  ft.  (HT.  sec. 

"'  _  li"!''  +15^) 

?'■  ~        TiiilT"  =  •-""*  '5'  +  '■'"  "  '^in  15-. 


•  n  —  cot  15"  — 


rosec    15 
1.972 


'•77277. 


and  „  _  ,^..  ^„.. 

iil   Hy  eq.  (i),  »,.-,  sin  y  --  ,,1,,  sin  /J  =  '*'  sin  ;9 

/■■ 

'"■  '*'"  /^  =(4)   *<    '•''72   X   .259  =  .79JJ04. 

and  n  =  52«  5f,'. 

(r)  The  energy  carrie<l  ,u\.iy  by  the  w,itei 

''»'       12;.     ,        ,ri        ir 


=  62S.1. 


64 


,jj4".    -Ill'  ,   ^     '//,Vi  _  cos /J) 


=     (54.09)"  X   •.•»97  =  145"  «  fi.-lhs. 
-  2.r,4  H.F». 
(rf)  Hy  eq.  (141,  the  loss  in  hydraidic  resistance 


'     ^'-i-'/   .7-. 


;l  II. I' 


Fx.rurt.FS. 


399 


(e)  The  total  possiljlo  work  =  62]  .  1  .  200  =  12.501)  ft. -lb 


The  useftil  work 


(/)    Ihc  cthcii: 


12500 


-    1451. H  —   1617.5  =  9430-7  ft. -lbs 
=  17.14'.  111'. 

=  .754. 


E.X.  2.  A  centrifugal  outward-ll*  uv  turbine  witli  an  ciricieiicy  of  Ro 
I)er  cent  ami  working'  under  tlie  liead  of  200  ft.  over  the  inlet  passes 
I  en.  ft.  of  water  per  second.  The  an^le  y  =  15°  ;  jr,  =  4/j ;  and  the 
velocity  at  outlet  is  radial,  i.e..  rt"  =  90°.  Find  (<»i  the  i)eripheral  jpeed  , 
iA)the  lip  angle  at  inlet  ;  (i )  the  ratio  of  the  inlet  to  the  outlet  depth  ; 
(</)  the  lip  an^dc  at  outlet  ;(*■)  the  energy  carried  away  by  the  water; 
(  /■)  the  useful  work.  (Disrej,'ard  the  blade  thickness  and  the  hydraulic 
resistance.) 


(<i) 


fiut 


V, 


64 


=  20(3.     .ind  therefore     T'l  =  8o|^2  =;  "3.17  ft.  per  sec 


.8  —  the  elhciencv  = 


iiiT'i  cos  15*       III  cos  15* 


(*) 


32    X    200  ^o\' 

u,    -  32  V  2  sec  13'  -=  46.851  ft.  per  sec. 
u,       sin  (<<  +  15  ) 


32V'2sec  15'      2 

^—^ =     sec  15 

8oV'2  5 


sin  <t 


and 


=  cos  I  5"  +  (lit  i*  sin  I  5°, 
cot  (180"  —  <t)  =  cot  15    —       sec  15*  cosec  15* 


:-    3.732050S    —    1.6  =    2.1320508. 

ind  <«=  154°  Si- 

(1)  By  cqs.  (10)  and  (121. 

M.'d   +  tan'  fi)  =   IV  =    f  ■.'  +  «i'  -  "■'  -  "-'  -^  -i'  -  2">''  •■•■'  •'• 
.r  «,'  tan'  fi  =  fj   ij]  ?•.'  sip'  15     -  :.■   -  ://m',  cos  15, 

)r  tt/tan'/i  =  8192  sin«  '5    [|/j   -  8oV'2(8of^2  -  64V2)  =  2560, 


which  gives 

and  therefore 
(,/)  Hv  ii}. 

2560 


=  2.16. 


tan'/<  = 


25(k)    _  2560 


cos'  1 5*  =  .7464. 


40O 


1:4 


'I'ii('ruf(;ri; 


anil 


ll-.T    lURBIK-E. 

tan  /i  :=  .864, 
n  --    40°  50'. 


If )  Tlif  energy  rarri(;(l  iiway  bv  t!ie  water 

r'j'        12!  12; 

=  62}  .  I  .    ,-    =       -  «,'  tan'  fi  =     -I  X  2560 
<>4        126  1 28         •" 

=  2:00  ft.-ibs. 

=  4A  11.  P. 

(/)  The  tutal  possible  work 

=  1. 62 J .  200 

=  12.500  ft.-lb.s. 

I  liMirc  the  useful  wnrk 

=  total  possible  work  —  2500 
=  12500  -  25'X}    -  10,000  ft. -lbs. 
=  i8,»,  H.l'. 

13.  Jet  Turbine.  — In  tlic  jul  turbine  tlu'  water  passes  alont; 
the  axis  ami  is  distrihutcii  radi.iily  in  ,ill  directions  so  that  tlie 
an.Ljlc  ;  _  yo  It  is  no  lon{jcr  possible  to  have  //,  =  f,  .  and 
it  cannot  tliereforc  be  assmiied  tliat  it.  =  l',.  A  fair  efficiency 
may,  however,  l)c  securcil  by  niakin^j  11,  —  Ty 


Fig.  233. 

Fifsf,   (lisrc^janl    hydraulic    resistances.      Mien,    Irom    the 
trian^Mc  it</i , 


JHT   TURIilNF.. 
and  2;\'  ~   l','. 

I'"r<>ni  till-  tnaiiL^Hc  fk/i, 

7'./  —  //,■  4"  '."'  —  -II  J',  (.OS  (i 


V-i  3—2*2  COS  fi). 


Hcnct 


///f  Hit  fill  U'l'lk 


^^•(J  :v  -  r/ 


401 


:.'0    '    I  2  \  2  cos  li  —   2). 


■  iiul 


///»■  I  nil  ii  II,  I         _M  J  CMS  li  —  2. 


or 


t  r 

llciicc,  too.  CO-  fi    ■         ,  i  I.,   /i  must  not  exceed  41;°. 
i  2 

Siidi/t/,  takiiii;  tlic  liyiir.iiilic  resist;ituc^  into  account, 
//,-  -t-  7-,^     .    ( •  ■■        (  I  4-y", '  / '.,-  -  11. r  -4-  «,-' 


Also,  the  Id---,  o)  hrad  up  t<>  inlet 

I'i  f  -);,-• 

"      in  \\  liiTl-p.issatres  —/'-=—'  ' 

ami  the  total  los-,  of  head  due  to  tile  /*/7//<7/(f/ hvdiaulic  iisist- 


anccs 


-IMM 


If  //  1-  the  he. id  ovir  tlir  iidit. 


//. 


XoTK,      InijJict.   ccntrifu>;al,   and    jet    turbines  will    work 
with  the  axis  iMciined  at  .im>  .m^de  to  tiie  vertical. 


^ 


fc:4 


402         RFsisT.-ixcf  TO  .Morim  ()/•  so/ ins  /v  /icins. 

14.   Resistance  to  the  Mrtion  of  Solids  in  a  Fluid  Mass. 

i  lu-    jirc\  ciliiiL;   result^  in<li   :.ti-    I'l  it  l!ir    ]iri'---un-  <lnr  to  tlu- 
impact  (if  a  jet  upon  a  surf  11  c  ina\    be  ex  tressed  in  tlie  torni 

/'• 


P  =  A;r. 


A  heiiiL,'-  the  seetimial  area  of  the  jet.  /'th(  \ei..iity  of  tlie  jet 
relatively  to  tin-  surface,  and  A' a  coeffiiient  liepenchiiL;  0:1  tile 
imsition  and  form  of  the  surface. 

A^ain,  the  mirmal  pressure  1  .\' 1  i>n  each  --iiie  ot  a  thi:; 
plate,  completely  sul)mer;_;ed  in  an  iiulefmitely  l.irL(e  mass  ot 
still  water,  is  the  same.  If  the  plate  is  made  to  mo\i'  hori- 
/ontall\-  uith  a  \eIocity  / ',  a  forv\ard  momentum  is  tle\elo])e(l 
in  the  watei-  iinmediat<-l\  in  h'ont  of  the  jtl.ite.  \\hile  the  plat' 
ti  nds  to  lea\e  hehiud  tile  water  at  the  hack.  A  jtortion  of  tlu- 
uater  carried  on  hy  the  plate  escapes  laterally  at  the  ed^e-- 
atid  is  alisorbi'd  in  the  ueij^hhor.iiL;  mass,  w  hile  the  re,L,ioii  it 
ori;4inal!)'  occupied  is  lilletl  up  with  otiu'r  particles  of  w.iter. 
Thus  till-  r.ormal  i)ressure  .\  .  in  trout  of  the  plati .  is  iucnased 
hy  an  amount  '/,  while  .it  the  hack  ed<lies  and  \ortic<s  arc 
l)roduci(l,  and  tlu'  normal  ]iri'ssurt'  .\' at  the  hack  is  diminislie<l 
l)\-  .ui  amount  //  .  Ilie  tot.il  resultant  normal  pressure,  or  the 
normal  ri'sistaui  i-  to  motion,  is  it  -\-  ii\  and  this  increases  \\  ith 
the  sjjeed.  In  fact,  as  the  s])eed  increase's,  ,"  approximates 
more  ;uul  more  closel)'  to  .\  ,  ami  in  the  limit  the  pressure  at 
the  hack  would  !h'  nil,  so  that  ,1  vacuum  im;^dtt  he  m.iintain<'<l. 

ConfMiini;  the  attention  to  ,1  pj.ite  movinj.;  in  a  direction 
normal  to  its  surface,  the  rc-sistanci'  is  of  the  same  chariicter 
a.  if  the  pkite  is  iinaL^ined  to  be  at  rest  a)i(l  the  thiid  movin- 
in  the  op])osite  direction  \\\\\\  .1  velocity  / '.  So,  if  both  the 
water  and  tlu-  ])!,iti'  .ni-  in  motion,  imai^iiu'  that  ,1  \elocit\- 
fcjuai  and  oppo'-iti-  to  that  of  the  water  is  impre-sed  u[)on  ever\' 
])article  of  the  ]>l.ite  atul  ot  the  water.  Tile  resistance  is  then 
of  the  same  character  as  th,it  'if  a  i)late  nio\iii^^  in  still  watci. 
tlu-   velocity  of  the    pki'e    beini;    the   \ilocitv  relativcK-   to   the 


KES/ST.tS'Cl-:    TO  MOllO.\   Oh   SOLIDS   J.\    I  l.LlDs 


■}oi 


water.  riui>^,  in  ^fciii.ial,  the  |■c•■^i'^t.^Kl■  to  tlic  iiuition  mI'siuIi 
a  plane  inovinj^  in  tlie  direction  of  iIr-  nornial  tn  its  surface, 
with  a  vcl<)cit>-  / '  relatnely  to  tlie  water,  may  be  expressed  in 
tile  form 


■A 


.1  1 


uinLj   tlie   area  of  tile   plate,  ,iiid  A' ;i  eoeffieient  -leiiendint 


upon  the  form  <.f  tlie  pl.ite  and  also  upon  t 
areas  of  the  plate  and  of  the  water  in  uhic 


he  rel,iti\  e  set  tional 


h  it 


is  Mibmerijed 


Accordin''  to  tl 


e   e.\])eriments   of  1  )ul)u.it,   Morin.   I'iohcrt, 


Did 


i(n<in. 


.M 


iriotle.  ,1111 


'Ihil 


ih.iiilt.  the  \  ,diK' 


<\'  A 


\  ma\-  he  t.ik 


ell 


.It  1.3  for  a  plate  movini;-  in  stil 


water,  ami  ,it  1 . S  for  a  current 


movmi/  on 


a  fixeii  plate.      I' 


nuin 


])oints  out  the  unlikclihi 


of  such  a  difference  between   the  t 


U()    WlIlK" 


.111(1  sii"'nsts  that 


It  nil 


;ht 


ibK 


he  due  to  errors  ot  nieasiiri'ment. 


A-ain.  reasonim.,'  Irom  ,iii,do;4>-,  the  resi-,taiicc  to  tlie 
motion  of  a  solid  b.)d\-  in  a  in.iss  of  water,  whether  the  body 
is  V  lioIl>-  or  only  parti. illy  immersed,  has  been  expressed  hy 
the  (ormiil.L 


A' 


A.-,.  I 


/■•• 


■; 


/■  heinc;  the  relati\e  \cl(uit_\-  of  the  body  .md  water.  .1  tlie 
};re.itest  sectional  area  ol  the  immersed  portion  ot  the  body  ,r 
rii,'ht  anL,des  to  the  direction  of  mntion,  and  A  ,1  coeflicient 
dei)eiuliiii;  upon  the  form  of  the  bodw  its  positi.n,  the  rel.itive 
sectional  areas  of  the  b.nly  aiiil  ..t"  the  mas-,  ,,r  water  111  which 
It  is  immersed,  .md  .dso  iijiiui  the  siirf.ict;  \'.  .i\  (.-motion. 
'l"he  (ollowiiiL;  \,dnes  ii,iv(.'  '.K.'en  i^iven  for  A": 

A' 1-  i.i  for  a  prism  with  plane  ends  .md  a  leii-th  from  thrc' 
to  si.x  times  the  least  transvi-r-.    dini.  iisiMii; 

A'-  i.o  for  a  prism,  ])lane  in  front,  but  tapeniiL;  tow.ird-  tiie 
stern,  the  curvature  of  the  surface  ch.myini;  -r.uhi.illy 


Ml 


w 


•^» 


404 


I'KHssuKf-:  ox  rt.-nn  i\  i^irr. 


so  that  tlK   ■^ticam-iiiu>  ^an  IImw  pa^t  u  iihout  aiix-  pro- 

iluctiim  I't'ciidy  iii.itioii,  (.'tc.  ; 
A'=^    .5    lor  a   prism    with   tapcnn-    stL-rn   and   a   cut-uatcr  or 

semicircular  prow  : 
A'  -^    .3s  for  a  prism  uitii  a  tapcrin-  stern  and   a   jirow   uitli  a 

plant-  front  iiKlmcd  at  30    to  tlic  liori/on  ; 
A  —    .16  lor  a  wcii-rormcd  sjiip. 

lM-(<iidc'>  oxpcrin)ciits,  liowcvrr.  sliow  that  tin-  rL-^istancL- 
to  the  motion  of  a  ship.  ,,r  of  a  hodv  tapering;  in  frnnt  and  in 
the  n.-ar.  so  that  there  is  no  abrupt  ch.  >  .  v  ol  curvature  Jeadin^r 
to  the  [n-oductioii  of  an  edd\-  motion,  is  almost  ciitire!\-  ^\\^v  to 
skin-friction  1  see  Art.    1.  Ciiap.   II   . 

15.  Pressure   of   a    Steady  Stream  in    a    Uniform    Pipe 

against  a  Thia  Plate  .{/:  Normal  to  the  Direction  of  Motion. 

-Tile    stream-lines   in   front   ot   tlu;   jilatc   aix   dcMatcd   ami   a 

contractir)ti   is  formed  at  C .C ,.      'l"he\-  tiien   conver-e.   ieavin^r 

a  mass  ot  eddies  behind  the  platf. 

Consider  the  mass  bounded  by  tiie  transverse  planes  C  C  , 
6  ,(",,  where  the  stream-lines  are  a^^ain  parallel. 

■•,  ■    .^1   be   the   mean    intensit\-   of  the 


At    (■.(",    let  /-, .    A. 

4  1  i        \  1 


pressure,  the  sectional  area  of  the 
uaterwa)-.  the  velocity  o|  |1,,\\,  .nid 
the  elevation  of  the  (.'.  oft;.  ,,f  the 
section  al)o\-e  datum. 

'•'■t    /•,,    ./_,.    ,-...    -^   be   corrc- 
-pondint;  symbols  at  C' X  '.,. 
-,  lie  correspondin.L;  symbols  at  C.C',. 

Let  ,/  be  tlie  area  of  the  |)late. 

Let  t\.  be  the  coel+icient  i>f  contraction. 

Nc<j!cct  the  skin  and  fluid  friction  between  (\(\  and  C  (' . 

Tlien,  by  neniouillis  theorem, 


■ 

wt  ^ 

i 

^t-^z 


~:  + 


A 


4.  !V  ^ 

1  ,-r 


A  ^  'i"    1    ^''i  ~_ 


•,)• 


PRESSURE  UN  rLAir  l.\  ril'l:. 


4-^5 


i^\-^'^ 


tlic   term    "  "-^'      rciircsL-ntinj,'   tlu-   I<.s^  <,f   lu-ad   i\w  to    the 

bcmlin_i,r  of  tlic  stRMin-luu  s  lictuccii  C„C,  .iiid  (  ,( 
Hciuc 


"I  ~'H       I  .  r . 

A^^aiii,   let  A'  In-  thr  total  i)rcssurc  dii  the  pl.iiu-.       'i  Jkii 

/.J,  -  /.J    =(/»    —  /-  U    ^^    *'^"'''    l"''-'^'''"'e    in    tlu-    Jircrtiuti 

I      «it"  the  axis, 


T..J/- d;-;,.^.-' =...../,(., -^j 


=  component  ..f  the   uei-ht  in  the  >lir.  etioi, 
of  the  axis. 


Thus 


(A  -A'./,  +  r.vi,(~,  --,)  - 


sinci--  the  nidtidt)  is  steady. 
1  ItiKi 


A'  -    chani^e  of  niotinn  in  direc- 
tion of  axis 

=  o, 


^'^      .  /.  -  A  , 


2g-        ■ 


lUii    J,r,  =  JjT'j  =  <vM,  -  ,,):,.       Iherefore 

A    —   Ji'j-l,         «    -     — — '- I   . 


'  I 


\        "I 


—  JiVJ  -   in  ■         —         —  I 


4o6 


PRESS Ufif-:    ON   CYLINDER   IN   PlI'E. 


wIhtc 


./, 


A'  =  K'u;i''  . 

If 


u  hero  A'  — 


/r,(w  —  I 


\ 


t^i 


i6.  Pressure  of  a  Steady  Stream  in  a  Uniform  Pipe  on  a 
Cylindrical  Body  about  Three  Diameters  in   Length. —  i  iic 

-trcmi-liiiL"^  in  trnni  "ftlK'  hudy  air  ikviatcd  ami  a  c<intr<icti<>n 
i-i  termed  at  C.C.-  llic)  tlun  I'liiu  lm  j^c,  tlnw  in  parallL-l  liiiL-., 
and  i.()n\cii^c  a  scrond  linu-  .it  (  X  ..  Icavini,'  a  nia--^  of  ci!die-> 
bcliiiul  tlu'  1)1  idy. 

Cniisidcr  tlu-  mass  Imundtil  i)\-  the  j)lanrs  (  \(  \  .   (  ,C',- 
A-i  in  tlic  ptcviiHis  article,  let 

/",  .  -i|,  ■"',  ,  .;•,  he  the  intensity  nl  piessiire,  seetiunal  area 
1)1"  the  \..iter\\a>',  velocit)-  <>t'  llou ,  and 
elevation    of   L.    of  (i.    abo\e    d.atuni    at 

/>.,  ./.,  ."  . .   ~,  he  similar  s_\iiil)oU 

^^  t"or('..r,. 

c,     ''         f..  .1..  r..  .:.  be  similar  symbol, 

I"l(-,.   2T5.  for    (  ',(  ',. 

/>,  .  .i|,   r,  ,    :,  \k-  similai-  syinbols  for  C/\. 

\ei;leet  the  skin  and  lluid  friction  betueen  (  \(\  and  ('/",. 

Then,  b\-  Hernouilli'>  theorem, 


t 


''  +  i  +  '^r 


r- 


I     ^  , 


/. 


+ 


A 


'•■.  -  -.v 


-:•  \i 


(:•,  -  ^\)' 


2A- 


bcms;    the    loss    of  lie, id    between   (',(',  .md  (T'.^   .  and 


'  '       --  ■  beui"  the  loss  of  head  between  C .(.  \  and  C ,C\- 
2C  ■•     1  I     » 


PRESSURH    ON   CYLINDER   IS    riPF. 


407 


H 


cncc 


A- A        i':i-   'V'    1    (■•:  -  -'^l 


+ 


2,C 


Hut 


.-^.r,  =^., 


A.  -a)  -A.. 


am 


.>J,  =  J,  -  ,1. 


Therefore 


--.+ 


A -A 


^ifL-W—  I  / 


./, 


ii\{A,-a)     A->i^   J 


S       »' 


VI 


i-i 


h\(in—t)      in  -  1) 


wlicrc  ;// 


ANo.  as  in  tlic  prccxdiiiL;'  article. 


v,  -/'.)-',-^- •'■■',* -,--.'  -  ^' 


V  =  O. 


Uiice 


A'   =  Wd    '    l» 


I  {m  -  I 


+ 


tir       I  I 


[»i  —  1 


=(,-.) 


■I 


^ 


w  lierc  ;;/ 


=  ^.  and 


/ )'// 


+ 


f>t^      n 


1  )-       {in  —  I  Y 


.(' 


-An 


^ 


Tliis  value  nf  A'  i-  al\\a_\s  less  tliaii  tin-  value  of  A'  for   the 


])hite   ill   the    j>r<- 


liiv,:   article   tor   the   s.uiie    va 


lues    of 


W,     (/, 


.mil  1' 


lleiue   the   pressure 


the   c\lin(ler   is   also  less   than    the 


:<jrresi)i>n(lin^  pressure  on  the  plate 


In  e\er\-  e.ise 


A' shouM  be  determined  by  experiment. 


■II 


40.S 


::x.-i:,u'i!:s. 


EXAMPLES. 


^ 


It  ts^ 


I.  A  stream  with  a  tiansver-c  smioii  ..f  24  <;q.   ins.  delivers  10  cu.   \ 
ft.  of  water  per  seL.m<l  a'i.-Mnvt  a  fl.it  vane  in  a  normal  direction,      h'uvi       ' 
the  prrssuic  on  the  vane.  ,/„^.    ,,,,j  „,^ 

::.  If  the  vane  in  example  1  moves  1,1  the  same  dircetion  as  the  im- 
pinj^'ins;  jet  witli  a  velocity  of  24  ft.  per  seeon.i.  lind  (,/)  tlie  pressure  an 
tiie  vine  ;  (/-)  t''e  tist-fiil  work  done;  (,  )  the  eirniem  y. 

.his.  (<i)  421^  ll)s.:  (/■]  10.125  f'  -l'»-;  ('f).2SS. 

3.  What  must  he  tin-  sp,-efl  of  the  vane  in  e.xamtile  2.  so  ili.u  the 
efTirienry  of  tlie  artanKement  m.iy  be  a  nhiximitni  .='  find  thi-  maxinuiin 
^■"""'••'"^-  .///.v.   20  ft.  per  see.:   j',. 

4.  I-iiid  ui)  the  pressure,  (fi,  the  useful  w..ik  done.  (,  1  the  eUieiencv, 
when,  instead  of  the  single  vane  in  example  2.  a  series  of  vanes  are 
introdiired  at  the  s.ime  point  in  tile  path  of  the  jvt  at  short  intrrvals. 

.h/.t.  1,1}  7o3Ji  ll)s.;  (fi,  16.875  fi.-il)S. ;  (, )  .4«. 
What  must  he  the  spe.d  of  the  vane  to  j;ive  a  maximum   <  tri.  ieiuy  ? 
Wh.it  will  he  the  maximum  iIFk  lency  .*  .7//^.   50  ft.  per  sec  •    - 

5.  .\  stream  of  watc-r  delivers  7500  ^jallons  per  mmiile  at  a  velocity  of 
15  ft.  per  second  and  strikes  an  indefinite  plane.  Kind  the  normal  pres- 
sure on  th  •  vane  when  the  stream  strikes  the  vane  (<j)  normally;  (/')  at 
an  anjjle  of  60°  to  the  normal.  ,/;,.,.  (.,)  585  g  lbs.;   2,, 2.,)  jhs. 

6.  A  railway  truck,  full  of  water,  inovirti;  at  the  rate  of  10  miles  an 
hour,  is  retarded  hy  a  jet  fl..wiMn  freely  from  ,m  orifice  2  ms.  sfpi.ire'iii 
the  front,  2  ft.  helnw  the  Muf.ice.      Find  the  retaidinjj  force, 

.-/'«.  7-97  lbs. 

7.  A  jet  of  water  of  48  sq.  ins.  sectional  area  deliveis  100  gallons  per 
sceund  aKainst  an  indelinite  plane  indinetl  at  30  to  the  ditertioii  of  i|„. 
jet  ;  find  the  total  [.ressme  on  the  plane,  neKlcclinj,'  frirtion.      Mow  will 

,1^    the  result  be  affi'cted  by  friction  .-  _./„,.   ~,^  Uj,, 

.  8,  If  the  plane  in  example  7  move  at  the  rate  of  24  ft.  per  secoml  in 
a  ilirection  ini  lined  at  60°  to  the  normal  to  tlie  plane,  find  the  useful 
Work  done  anil  the  efTiciencv.  ./«,.  2250  ft. -lbs  •    ', 

At  what  an^Jle  should  the   jet  strike  the  plane  so  that   the  elhrMnc'. 
inixht  be  a  maximum  ?     I'ind  the  maximum  eiriciency. 

.I.1S.  sin-'  J  ;  j. 
9.  A  stream  of  32  s(|  ins.  sectional  area  delivers  r{  cu.  ft.  of  water 
per  second.  At  short  intervals  .1  series  ..f  tl.it  vanes  are  introduced  ai 
the  same  point  in  the  path  of  the  stream.  At  the  instant  of  impact  the 
direction  of  the  jet  is  at  riKlit  angles  to  the  vane,  and  the  v.ine  itself 
moves  in  a  direttion  inclined  at  45'  to  the  noinial  to  the  \ane.     Kind 


t:\.iMru;s. 


409 


llic  -.pci'd  ■>!  ilii-  v.uu-  will,  li  will  tn.iki-  the  cirii  iciicy  a  ina.xiiiiiim.     Also 
liiiil  llu;  in.ixiiiiuiii  liliciciii  y  aiul  I  lu-  usi'lul  work  (li)tii'. 

.I'li.  15.08  II.  per.  sec;  /.  ;  iioOjJ'i  (t.-lbs. 
10.    .\  stieam  nf  water  ol  1  sq.  (l.  sectional  area  delivers  Id  cii.  II.  per 
second  iKjrmally  aj^auist  a  tl  it  vane.      Find  the  pressure  t,n  the  vane. 
If  tlie  vatie  nxjves  in  llie  same  direction  as  tiv  inipiiii;in^  jet,  with  a 

on   the  v.iiie  ;  i/'i  the 


/  velocity  of   3-  ft-  IXT  sec.)!!'!,  tirnl    (./)  the  pres 
useful  work  done  ;  (.1  the  etFiciency 


How  woiihl  th'.'  results  be  alTected  if  the  vane  were  inclined  at  45    to 


ihe    let, 


id  move 


ill  the  direction   of   its  normal   with  a   velocity  of 


J4  ft.  per  second 


.his.   4000  lbs.;   22",o  lbs.,  72,cxxj  ft. -lbs. 


1S02.8  lbs.,  43.26S 


1;  -llx..  .K.4. 


lie  feet  of  water  ; 


disch.i 


r};ed  per  se( 


ind 


under  a  pres- 


ume I'l  too  lb>  per  sq.  in.  thronyh  a  tlun-lipped  onlice  in  the  vortical 
side  of  ,1  vessel,  and  strike  u^aiiist  a  vertical  plate.  Find  the  pressure 
on  the  |)late  and  the  reaction  0:1  the  vessel.  .1)11.  475. S;  lbs, 

1.:.  .\  st.^-'ain  moving  with  a  velocity  of  16  ft.  per  second  in  the  di- 
rection .1/,'C  strik'-s  obliquely  aijaiiist  a  flat  vane  an<l  drives  it  with  a 
vlocityof  X  ft.  per  second  in  the  direction  /)'/>,  the  anyle  ( ///>  beinj; 
•,o  ,  Find  (a)  llie  anj{le  belwoen  W/>'C'anii  llie  normal  lo  the  plane  for 
which  the  cfliciencv  is  a  ina.\iinum;  (/'l  the  inaxinnini  efficiency  ;  u)  the 
velocity  with  which  the  water  leaves  the  vane;  (1/)  the  useful  w<-rk  per 
ciibii'  foot  of  water. 

.Ins.  (,i)  21°  44';  (A)  .25664  ;  (,  I  12.6  ft.  per  sec:  (./)  256,64  ft. -lbs. 

1  V  At  S  knots  an  hour  the  resist, nice  of  the  Water-witch  was  5500 
lbs.;  till'  two  orifices  ((f  her  jet  pri>pellcr  were  e..ch  iS  ins.  by  24  ins. 
Find  (<il  tlie  velocity  of  efflux  ;  (/•!  the  delii'eiy  of  the  rentrifiij;;d  piinip; 
((I  tlie  useful  work  done;  (./)  the  etTi<  iency  ;  i/'i  the  propelling  II. P.. 
assuming  the  eliic  iency  of  the  pump  and  engine  lo  be  .4. 

.Ins.  (.»)  20.4  ft.  per  >e«-.;  (/')  il  14.6  ^jalloiis  per  sec;  (.  1  74..193 
ft. -lbs.;  (./»  .63;  {,)  536,7. 

14,  If  fi'atberin^;-paddles  ure  substituted  for  the  jet  propeller  in 
()ueKtion  15,  what  would  be  tin-  area  of  stream  driven  back  for  a  slip  of 
25*.*     Find  the  efTiciency  and  the  water  acte<l  on  in  (»allon5  per  niiniite. 

.hts.  3463  !<q.  fi.;  .75:  234:0^., 

15.  \  let  issues  hori/ontally,  under  ;;  head  c)f  20  ft.,  from  a  i-in.  ori- 
fice in  tiie  veriical  f,ae  of  a  i.ink  .ind  strikes  normally  the  centre  of  ,1 
v.uic  at  a  ilistance  of  48  ins.,  measuied  lior>/oiit,illy,  from  the  t. ink's  face 
Hv  nieasurement  the  verticil  di!tt,ince  of  the  point  of  impucl,  bdow  die 
axis  of  the  orifice,  was  found  to  tie  2.5.S2  inches.  Find  the  ctH'tl'icient  of 
velocity  <!,),  the  inclination  of  the  vane's  nxiR  to  the  horizontal,  and  the 
coelFlcJent  of  impact  (<,l  in  the  follov   n^  cases  : 

(>n  A    flat    li-iii.  circular   vane,  the    balancing    weight    {It')   Ixnig 

3.015  1I)S. 


51 


iJi 


ill 


.(lO 


l:.\.-l  UP/  /:.V. 


If.' 


m 

,  1 


lit 


4  * 


'^ 


N 


li)   A  hi'inisiihcricM!  v.iiic  of  i;  ms.  (ii.imriri.    II 
(i)   A  liLMiispiicric.il  Viiiie  of  3  ms.    h.iim-lc:.   //' 
(</)  A  |i,irabolic  v.iiic  wilh  a  base  ot   12  ins.  Jii  d 
iu'i;;lit.   "' beiiif;  3.535  lbs. 

{.)  An  clUptic  vane,  (j   ins.    in   Iici^^lit  and   liavinj;  a   b.ise  of    12    ins. 
3.56  lbs.     Tbf  vane  edj^e  is  iiiclii.fd  at   20"  tn  the 


■  bi-in^r  3,556  lbs. 
bciny  5.776  lbs. 
i.uiu'ti  r  and  6  ins.  in 


•■)   .'MIO;    u/)    ,6o.S6; 


diarnctfr,    II'  bcin 
axis. 

Ans.    .'/)4ii  ;  6"  8';  i,i)  .9834;  (/')  .5799; 
W.59«^>. 

16.  Find  the  atiyle  of  l,..idi'  at  ciitrai-.rc,  the  iij^olul  cIIl-i  t,  .ind  the 
elfiiienry  i)f  ,1  Hor'l.i  turbine  fnun  tbc  foll.ivvniL;  d.ita  :  tlie  jet  at  eiitraiu  e 
makes  an  ant;li' ui  ()0  with  tlie  linn/.onlal  ;  the  deptli  of  thotiiibme  Is 
3  It.;  the  total  fall  to  the  point  of  discliarj^e  is  Ij  ft  ,  thi-  nir.in  d:a;neur 
of  the  turbine  is  4  ft.:  tlic  (juai'tity  of  water  p.i-.sin);  thioiij;h  the  tni- 
bine  is  4CU.  ft.  per  second  ;  tlie  aM<;le  of  bl.ide  at  exit  is  3o^  (I)isref;,(id 
hydraidir  resistance.)  his.  51°  33';  7.256  H.l'.;  .84. 

17.  Wli.it  , idv. images  arc  g.iined  by  increasihfj  the  depth  and  diaiii- 
ftcr  of  a  H.iid.i  luibine  and  by  cuiving  the  outlet  lips  of  tlie  buLkcts  .' 

18.  A  Horda  lurhinc  of  3.5  ft.  mean  diameter  lias  a  head  of  12.96  ft. 
over  the  inlet,  a  practical  eiriciency  of  .75.  a  theuretic  eHi(  lencv  (i.e.dis- 
rcjiardini,'  hvihatiiic  resistances)  of  .926;  and  delivers  3  horse-p.)wer.  The 
radial  width  ..I  the  w.ilcr-pas.sanes  is  3  ins.,  and  the  depth  of  the  tuibine 
is  2.04  ft.  If  tlKTe  ir.  to  be  no  shock  at  entrance.  Ilnd  (,i)  the  inln  .mil 
outlet  lip  angles  ;  {/>)  the  velociiv  (Tj)  of  discharge;  (.)  tbc  (piaiitilv  of 
w.iter  used  by  the  turbine. 

./«>.    (.»)  Ill     25'.  Jj     iJ   ;  .,«)8.4    ft.  per  see;   (,)    2.532  cu.   ft. 
I)pr  fee. 

19.  In  a  railwav  triuk.  full  of  water,  an  oi>piiiiij;  2  ins.  m  rliamcter  is 
made  in  one  nf  the  ends  of  the  truck,  9  ft.  below  the  surface  ot  the 
water.  Find  the  rc.iction  i<i)  when  the  truck  i.s  stan(,inn;  ((4)  when  the 
truck  is  movinj;  at  the  rate  of  10  ft.  per  second  in  the  same  direction  as 
the  jet;  U)  when  the  truck  is  m'>viiii{  at  the  rate  of  10  ft.  per  second  in 
a  direction  opposite  to  tbat  of  the  jet.  If  this  iii.>vement  of  the  trin  k 
IS  proilui  ed  by  the  reaction  of  the  jet.  lind  the  elfiiiency. 

./«>.    {,,)    24.55    '""•  K'r    "•I-  '"•:    »'''•    34-r«  lbs.   per  sij.    in,; 
((  )    I  ',.3  lb*.  |)er  sq.  in.;  .588. 

20.  From  a  ship  moving  (orwarcl  at  f>  miles  an  hour  a  jet  of  water  is 
sent  astern  with  a  velociiv  relative  to  the  ship  of  ^o  ft.  jwr  second  from 
a  no/zle  havini;  an  aret  of  16  »(j.  ins.;  find  the  prop  Ibm;  force  and  the 
etrui.-ncy  nf  the  jet  as  a  propeller  witlnul  relereiu  e  to  the  mannei  m 
wliii  h  (he  supply  of  water  m.iy  be  obtained.         .-/«.«.  1  ?8,',  ibs  ;  .4535, 

21.  A  reaction  wheel  is  inverted  and  worked  as  a  pump.  Find  the 
sjiccil  of  m.ixiiiinin  cIFk  lemy  and  the  niaximuin  eth<  ieiicy.  tlic  coeffi- 
cient of  hydraulic  resistance  referred  tu  the  orifices  being  .ijj. 

.!«<.  Spee<t  =  twice  that  dii.-  to  lift;  .7j8. 


II 


i<n 


f:\.-iMri.HS. 


411 


A  react i 


,  lii-rl  with  nnrir( 


111  (li.iTnetcr  ma 


kes  So  re\  ')hi- 


tiiins  \i 


.T  mi'uilc-  unilrr  a  Iumi 


1  of  s  II. 


ilotai 


III  an  ..iilu:i'  ami  llic  axis 
the  ciru  ici 

23.   In  a  reaclloii  wlic 


jf   iDtaliuii   IS  i  J   11 


■  iH-lwcfii  ihe  centre 
I'ind  tl'.e   U.V.  and 


.■/«v.  .14'' 


.3'/' 


el  the  speed  uf  maxiinuni  ciricu-m  y  i-.   thai  due 


It)  the  head.     In  what  r.uii)  must  the  resistanc 
at  ^  this  speed,  and  what  will  then   be 


hi'   dimiiii--hcd   t<i  worl; 


the 


■Ihc 


;v  ?     <  )bt.iin  sinii 


hir 


results   when    tlie   speeil   is 


(Imiiiiishen 


til  three   f.iurths  o 


f   Its 


)rii;inal 


amiiiiMl. 


./' 


')A. 


.KSt/) :  1.067 


24.   In  a 


reaction  wheel,  determine  tiie  \)ci  cent 


1 1  ava 


lable  ciTect  lost 


(I)  i( 


J'-  -  :(,'//;  c;! 


if  f^  -  -U'//;  (3)  if  /■"-- «<.'//• 


Wliat  conclusion  may  t)C  (Ir.iwii  tr 

Ktn 


the  results ; 


cieiicies  arc  resp<'Ctively  .X-S.  .9.  .<M5- 


:;.   .X  stream  of  64  s<(.  in*,  set 
a  fixed  cone  luiving  an  aiiyle  of 


section  strikes  wilii  a  40-ft.  velocitv  against 


coDverjjence  =  100 


j)ressure. 


jfi.   A  jet  of  0  sq.  ins.  sectional  are 


liiid  the  hvdrauli 
.//;«.  4<)2.i  lbs. 
in^'  at  llie  rate  of  48  It.  per 


second.  iiiipinRes  111 


1)  the  convex  surface  of  .1  paiaboloid  in  tin-  direc- 


tion of  the  axis  and  drives  it  111 


the 


line  direction   ,it  the   i.iti 


.f  16  ft. 


[«■ 


d.      Fiii'l  llie  foiie  in  the  direction  of  motion,  llir   n-iliil  worl, 


lone,  and  the  elh<  iency.       Tlic  base  of  the  paraboloid  is 


ft.  in  dianiet<r 


d  lis  ;on(»lh  is  S 


./;;.( 


bs. ;  4<w  ft. -lbs. ; 


.•\  stream  of  water  o 


)f  l6sq.  ins.  .sectional  areadeliver>  12  en,  ft.  of 


f. 


rm   ol   a  siirt.u 


if  revolution. 


water  i)er  second  against  a  vane  m  tin' 
and  drives  ill  tlie  same  <lire(t  ion.  v.  ii..  h  1-.  lli.it  of  tli.  .i.\is  of  the  vane. 
The  w.iter  is  turned  lhrouj;h  ,ni  .iiif^le  of  60  from  ts  oti-inal  direction 
before  It  le.ivcs  the  v. me.  NeKleciin:^  fri. lion,  find  the  speed  of  ^.mr 
which  will  yiw  .1  iiLiximnm  elTe.  t.  .\;.o  hiid  impulse  on  vane,  the 
work  on  vane,  and  tin-  >elocitv  with  \\hM  li  the  w.iter  le.iv.s  the  v.in.' 
Ins.  }(<  It.  per  sec.  ;  5621  lbs.;  20.250  (t.-lbs.  ;  ^5.24  "■  I"  r.  sc<  ■ 
2K.  A  surface  of  revolution  i-  driven  in  the  direction  of  it>  .ixiswiili 
.1  velocity  of  16  ft.  irt-r  second  bv  im  ni.  ..f  a  in  of  wain  of  iS  sq.  inv 
K-cti.in.il  area,  which  moves  in  the  direitioii  ol  tiie  .ixis  with  a  Veloc  ilv 
of  80  ft.  per  second,  and  impinges  upon  the  convex  sidi-  of  liie  surfaie. 
The  tangent  at  the  edj;e  of  the  surfai c  makes  .in  an^;le  of  30°  with  ti.c- 
verticil.      Find  the  piessure  on  the  ^urf  i>e  ..nd  the  ethciency. 

W«t.    500  lbs.  ;   .128. 
29.   .\  jet  of  w.itri    under  .1  liiad   ol    Jo  It.,  issuing   from    a  vtrMcd 
t'.iiiidipi.ed  orifice  1  in.  in  diameter,  impinges  upon  the  centre  of  .1  v.im 
\  ft.  ftom  the  onhce.      Determine  the  position  of  the  vane  .iiid  tlie  lone 
01  the  impact  (.1)  when  the  vane  is  a  pl.ine  surface;  (.*)  when  the  vane  is 
6  in«.  in  diameter  and  m  tin-  l-mi  ..1  .1  ponion  of  .,  sphere  .>f  ''  in--. 

radius, 

./«.t.   (.11  I.V')7<»  lbs.  ;  (/i)  30.518  lbs.  or  6.839   'l^'*-   according!   as 
\Ai\K  is  concave  or  convex. 


in 


■f 


41. 


-':'.V.-/,V//7/>. 


II  III 

mil 


jo.  A  -iuMiii  .,f  water  1  m,  th,,  k  ..n.i  ,S  nis.  u  i,l,-,  in-vi,„.  vvilh  a 
velonty  of  iS  ft.  |„.|'  second,  strikes  wuh,,iit  >|„„k  a  <  ii.iil.ir  v'.im-  of  i 
l.-ii.ulli  siihtcnrlmn  an  angle  of  <;o°  at  the  ceiitr,..  1  ,„•  vanr  is  ,i,iv, 
the  (lirei-tioM  ol  tne  Mrr.iiii  with  a  velocity  ,,i  (,  u,  p,.,  .orojid  1 
the  prcs-ii,,.  ,,:,  the  v.ine.  the  w,ul<  ,l„ne,  and  the  elVuien.y. 

■  his.    22,,'.,  Ib.s.  :  95'j  ft.-lh>.  .   ,,-.. 

;,i.  A  I'rilon  wlieel  of  2  ft.  .hanutei  inaki  s  822  revolutions  per  min- 
ute uiui.ra  |.iessi,re-hea(l  of  2o<j  11,..  p.r  Mpiare  inch,  the  delivery  ol 
w.iter  being  i^x,  ,  ,,.  it,  pr,  niniiite.  I'm,!  the  t-lal  ii.l'.,  assuming  'that 
the  Inickets  .ire  s,,  |,,irne.i  tli.it  tlir  UMt.-r  is  retuine.l  par.illel  to  its'.rigi- 
nal  direction,  ami  i(  ,ivi-s  uitii..ut  mcryv. 

If  The  a,  lu.,1  111',   :.  70.;;.  wh.it  is  tlie  .truienry  .- 

.Ins.  .S7.22  ;  ..S05. 
32.  A  vane  iiu.ves  111  the  dirertion  . /At '  with  a  vclocilv  of  10  ft.  per 
se.otui.  and  a  jet  of  water  impinges  i!i>on  it  at /.' in  the  dii  .tion  ///> 
with  a  velocity  ol  20  ft.  per  second;  ih--  angle  between  /.'t  and  /y/>  is 
30°.  Determine  the  (iiicetioii  o(  tin-  re<  eivingiip  of  th.'  vane,  ,0  that 
there  in.iy  be  no  sIkk  k. 

.■his.    The  angle  between  lip  and  />'C  =  2^  47'. 

ii.  A  jet  mov.-s  ,n  a  direction  ^Atwitn  a  vel.H  itv  T' and  impinges 
upon  a  vane  win,  h  it  <lrives  in  the  d.rertion  /,'/)  with  ,1  velocity  'v 
The  angle  .//,•/)  ,s  |.,5'.  Determine  the  direction  ,,f  ilu-  lip  ,,f  thr'v.mj 
at  />',  so  that  tlKTe  may  be  no  shock  at  cntr.inci-, 

.■his.  The  angle  between  lip  and  direction  of  stream  -  14^  3'. 

34.  The  lip  angle  of  a  given  bucket  is  ',0°,  the  rel.itivc  \eh)city  ( /• )  is 
one  half  the  vc-l<«  ity  (7.,)  with  which  the  water  reaches  the  lip.  If  there 
IS  to  be  no  ••  loss  in  sIi.k  k."  lind  the  speed  l,t)  of  the  bucket,  the  direc- 
tion (r )  of  the  .ntei  ing  wal<  r.  and  show  that  if  the  speed  is  to  be  increased 
10  per  cent,  the  hp  .ingl,-  must  ,ds<)  be  mcreaserl  by  55  6  |)er  cent, 

.///.-.  y  -  1^-  51'. 

3>.  A  slream  moving  with  a  velocity  ;•  impinges  with..ut  sh.Mk 
upon  a  curved  vane  .ird  drives  it  in  a  direction  in<  liiie.j  at  an  an-le  to 
the  direction  of  ihe  stream.  The  angle  b.^tween  the  lip  of  J.e  vane  and 
me  direction  of  the  stream  is  .r,  and  fis  the  relative  velocity  of  the 
W.iter  with  respe,  t  to  tiie  vane  If  the  spee.l  of  the  vane  is  changed  by 
a  small  amount,  say  //  p,-r  cent,  show  that  the  .  orresponding  ch  ingc  in 
the  directum  of  the  lip,  in  order  that  the  water  might  Mill  strike  the 
vane  without  sli.ick 


IS    -    ,.  sin  X. 
lou  f 


]]. 


36.  A  jel  issues  through  a  thin-lipped  oriike  I  sq.  in.  in  sectional 
area  in  the  verticil  side  of  a  vessel  uiid.r  .1  pressure  equivalent  (..  , 
hea.l  of  .)oo  ft.  ami  impinges  on  a  curved  v.me,  driving  it  in  the  dire.  - 
mm  of  the  axis  of  the  jet.  The  water  enteis  without  shock  and  turns 
through  an  angle  of  f.o'  before  it  leaves  I'e  v.iiic.  r-iii.l  (,/)  the  spi-ed 
of  the  vane  which  will  give  a  maximiini  efi.ci  ;  (M  the  pressure  ,„,  the 


;:|.  n 


11 


EX.tWI.liS. 


413 


vane  ;  U)  the  work  lioiic  .  i,/i  ;ln-  .ib^uluu-  ^ducity  ivitli  uiiicli  the  water 
leaves  I  lie  \arie  .   (,  )  I  lie  reai  tiDii  mm  tlie  vessel,  disrcfjaidini;  ((intra'  tion. 
./;/v.  (,()  «o  tt.  per  sec.  ,  {/>)  320.9  lbs.  ;  (c)  46.6.S  11.1'..  (,/)  184  ft. 
[ler  see.  ;  Um  7.S1. jj  lbs. 
j~.    A  stream  of  thickness  /  ami    moving'    with    the   vcI.k  ity   .'    im- 
pinges witiiout  ~.\\'nk  11)1. .11  the  .  .jii.  ,ive  surlari;  ot   .1  cvlm.lne.il  vane  ct 
a  ien{,'til     subleinling  an   angle  2,1:  at   llu-  OMiiie.      Deleimine    the   total 
inessure  npon  the  vane  (,n  if  it  i^  lixeci  ;  (''.  if  it   is   niovint;   in   the  same 
rlircetioii  as  tlie  sire.im   Willi   the  vel.n  ity  /'.      in  (  ,ise  ■'\  ,il,o  lind  (c)  the 
work  (lone  on  the  v.irie. 


r/)'  sin  If 


ID    sin'  n. 


I   2     /'0<[:- 
K 

•ecti'iiial   .lie, I,  deiivers   1   i_u.   ft. 
I   a   heinispheru  al    cup,    wliieli 


./wf.  (.')  2     /'/:■'  sin  ,r  .  I. 'I  2     /-A: 
.».'  A' 

J.S.    .\  siieam..]    watei.   2    ^.|,    m--.    in 
per   second  against    tlie  loncave  side 
moves  with  ,1  velocity  of  20  ft.  per  sec'iid  in  the  direction  of  the  )et. 

I'"in(l  tile  impulse,  the  work  done,  and  tin-  etficieiicy. 

y).  .V  (  !irv.  (I  v.iiie  -Mildends  ,111  angle  ..I  .«'  at  the  point  ot  intcisec- 
lioil  of  ill.  norm, lis  al  tiie  iwo  eds^es,  .md  receives  without  siio<  k  a 
stream  of  water  2  ft.  wide  .iiid  I  m.  tl:i(  k,  niovini^  with  a  vel.i(i(v..l 
ao  It,  pel  .second  and  dnvmu  the  v.inc  in  tin  same  direction.  'Ih  ■ 
actii„l  direction  of  the  water  i^  tin  11. ■. I  through  an  angle  of  45°,  I'md 
{a)  the  speed  of  the  v.ine  ,  ^h)  the  \cloi  ay  with  which  the  water  leavt  s 
llle  vane  ;  (i )  the  tot.il  pressure  on  the  v  ine  .  1,/)  i  he  .liii  lem  y. 

.Ins.  ill]  10  ft.  per  sec.  ;  (/-)  14.14  ft.  per  sei  , ,  u)  23,017  lbs.;  (,/)  .ij. 

40.  .\  vane  is  m  the  form  ol  the  s.-i;n.-nt  .//>'  of  a  circle  subf  ndim,' 
an  angle  of  1 20' ,it  the  centre  O.  .\  stream  o'  w.itii .  moving  with  ,1 
VehKity  7',  .  strikes  the  vane  langentially  a'.  .1  and  drive-  it  m  the  same 
flirectioii  with  ,1  \(  I..1  itv  //,  I'liid  the  velocity  (7',)  vvilti  whi.  h  the  w.iter 
leaves  the  vane,  and  sIk.w  ih,i!  it  le.ivi-s  in  the  direction  lUi  if  7.,  —  iA«, 
and  that  the  direitK.ii  h.iN  i::it;c.|  tlii.iiyh  go'  if  r,  =  3«,  Find  the 
ctTlciein  y  ill  the  two  c.ises.  an. I  >li  .w  tli.i 
mum  etricieiicy. 

.  /;/v   V,'  -  7','  -  yv.u  +  11';   jj  ;  ♦.     If  v,  =  2,1,  :;  =  /,.  the  direr- 
lioii  tiiins  through  (x)'  and  »/   -  J. 

41.  .\  SI  ream  of  water  of  36  stj.  ins.  sc,  tmn  ni,.v.',  w,  ,,  direction  ./Ac" 
.ind  deliveis  4  <  u.  ft.  of  water  per  second  upon  ,1  vane  moving  i"  , 
direction  IH>  with  u  velcKity  of  «  ft.  per  secoml,  the  angle  between  /.'t 
and  H/>  being  30^  Tind  (di  the  best  |..ini  to  give  to  the  vane  ;  {fi)  the 
velocity  of  the  water  as  it  leaves  the  vane;  1,1  the  mechanical  ellect  of 
the  impinging  jet;  audi,/)  the  elli,  lem  y,  the  angle  turned  lliLai^li  In 
the  iet  being  ">j', 

Ans.  (a)  The  angle  briwccn  lip  and  HC  —  23*  48' ;  (/)  3.088  ft. 
per  second  ;  d)  (;0»,8  ft.  ,»er  second  .  [,h  .963. 

42.  Ill  an  I.  V.  tangential  turhiiie  lind  (,i)  the  los.s  due  to  hvdrauiic 
resistances.  '*)  the  useful  ertect.  (>  i  the  .  Ificienty,  \J)  the  lip  mgles  from 


yt  I  .irresp.inds  to   ni.i.Ti 


'>. 


414 


i:\-iMri.H: 


i=i"    ?!'. 


1^    \^ 


tlu:  f.)ll,)uin.-  liat.i  :   n    =  i   ,ii.  ft.  p.r  second  ;  //    -_   rcxj  ft.;  /,  -^ /, 
y  -  \-,-  ■  ;, ,    -^  („;  ;  ,/,  =  ,/.. ;   ;,„,i  „,,  ^   /  -^ 

■  l"s.  (./)  736.72  ft.-!bs.:  (fi)4>ih(K>)h.\bs.:  u).;7,S7  ■  , 
'■■■       4:°  I'''. 

43-  111  an  ().  I'  1 11, 1, IMC  (;f  ilic  t.,ii-,-.ui.ii  tvpe  li.ul  ilie  Ii[.  iingl,'^  tlif 
losses  of  head  <hu-  v,  the  velocity  (r-.,)  of  the  effluent  w.iter,  aii.l  to  hy- 
draiilie  re.sistane.s.  and  also  hnd  tlie  eliiciency  from  the  foliowin.'  data  • 
y  =  30°:  2> ,  =  , ., :  //  ^  30  ft. :  ,A      ,/, ;  /,.,  =.  r,;  /;  ^  2/.  =  .1 25." 

.his.  a  ^  123°  27':  A  -  13^  30';    r.r.SS  ft.;   5.243  ft.;  .769. 

4t.  An  I.  [•.  tai)}.ential  turhine.  with  parallel  faces  (,/,  -  ,/,)  and  an 
inlet  siiifaerof  r,  ft.  diameter,  delivers  iu.7^.  H.l'.  u.KJer  a  head  .,f  150  ft. 
The  direction  of  the  il>t^nvin^r  stream,  which  is  5  ,ns.  wide,  makes  ati 
an<;le  (yt  of  10'  with  the  tnrl.ine's  periphery,  and  th,'  diameters  ,,f  the 
inlet  and  outlet  surfaces  are  in  th.'  ratio  of  4  to  ;.  If  /:,  =  ^  =  .1,  atul 
if  also  It  is  assumed  that//,  ^  C.lin.l  (.0  the  inlet  and  oiiii.;'t  lip  .mVies 
f^)the  loss  of  II, I',  due  to  hydraulic  resistances;  (<  1  the  loss  of  H  1'" 
correspondm-  to  the  velo.ity  with  which  the  water  leaves  the  turiiine  ; 
I./)  the  efli,  icncy  :  (,-,tlie  (luanlity  of  water  passing  through  the  turbine  i 
f/i  the  thickness  of  the  intlowm;.:  stream:  f.-l  the  speed  in  revoliii  i.„,  ' 
per  minuti- 

.hlS.i.l,  lOl  13',  40  30';  |,i)|.30():  (,).70.S;  (,/).,S42;  (<•)],,!. 
ft.  per  second  ;  (/j  .231  in.  ;  (  ^')  141. 
45-  111  llie  preceding;  e.xamiile  if.  instead  of  makini;  //,  ^  t\  .  it  is 
■  ■■sumed  that  the  (low  at  outlet  is  ladi.d.  tind  the  inlet  ami  ouVlel  lip 
an-ies  s  >  tiiat  the  elhcieiicv  may  remain  the  same.  Also  find  thi-  lo.s^ts 
111  H.l'.  du.-to  hydr.uilic  resistance  and  to  the  ener-;v  carried  avv.iy  l,y 
tiieetlluenl  water,  a). d  determine  llie  speed  in  revolutions  per  minute. 
"'"''•*  ■"'"*•    I'"*  21',  35°  17' :    1.360:  .(^,23  ;    139.S. 

46.  A  .stream  4  ins.  wide  and  supplyinj;  }  cu.  ft.  of  v,  ,t,r  per  second 
drives  an  ().  F.  turbine  of  the  t.int-ential  type,  in  whu  h  the  diameters 
of  the  inlet  and  outlet  surfaces  are  in  the  ratio  of  3  to  4.  The  turhine 
faces  arc  parallel,  .md  the  inflowing  stream  makes  ,in  aiifjle  of  20°  wi:li 
the  pcripherv.  The  head  is  100  ft.  /■»,/  assiimini,'  that  //.,  =  / ', ,  .uul 
sfuin.i  tli„t  the  outlet  How  is  radial,  the  eliiciency  iieiii);  the  same,  de- 
teimine  (,i)  the  inlet  an<l  outlet  lip  an-les  ;  (/-)  the  us.ful  work;  (,)' the 
.thciency;  (,/)  the  thickness  by  the  stream;  (ei  the  speed  In  le^ou- 
tlons  [ler  minute,  the  outer  diameter  bein-  5  ft.  (Disiegard  liv<ii.iiil  c 
resistances.) 

^ns.     First.      (,i)   140-.    2|-   |V;    «*)    436.S.16  ft.-lhs.  :    (,)    .<)',2  ; 
('0  -3.^75  !>'■;  <*)  216.7.     .Sfiond.     (,o  142°  19',  21"  10';  (/-)  4394.7 
ft.-lbs.;  (,)  .937;  (</).3375w;  (c)  202.45. 
47.  Solve  the  pieiediMK  example  when  hydraulic  resi.nances  are  i.ik.ti 
into  account,  assiiinin).{/,  =/,  -  .1  .md  ,■,'  =.-  .9. 

Afis.    lir.t.      (.1)    140°.    21*     u':    (h)   3766    ft.-lbs.;    (,-,    .S034  ; 


EXAMPLES. 


41^ 


id)  .35C.    in.;   ('I    I54. 


St'Ciiii. 


J.     (. 


n    141"    2'/,  20'    40';  (fi)  i'i.C 


out 


It.  lbs.;  (( )  .S034 ;  (li)  .356  in.;  (fi  147.79- 
4(S.   A  jft  turbiiic.  of  5,',  (t.  exterior  fliametcr  .nul  with  eiiual  iiiIli  ami 
111   (liptlis,   passes  1S90  cu.   It.  of   water  per  li^iur  iiiHicr  a  heail  <»l 


ft.;   tl 


le  (liainctcr  o(   the  nutlet   surlate  is 


twice  that  of  the 


and   llu;    velocity  of   the  outlet   [leripheiy  {:i,)   is  equal  to   tiial   of   tli 


iiitlowm^  Nttcam 
elli-ft  m  li<  .TM    pow 
minute, ///>/  disre^ 
resistances  into  aci Diiiit.     ( 


(7',),  which  is  radial  in  ciinctinii.      l-'md   t.t)   the  usctul 
}wer  ;  (/)  the  etri<;iencv 


dins. 


it  I  tlie  ^perd  lii  ici.  ol ill  ions  per 
Mlraulic  resislaiiies  and  Sc\oiiJ  tal^iii;,'  these 
/.  =  2/,  =.  .2  and  <•,■-•  =  \'\.) 

.  I  It -./■'/•  s/.    (.M  3.S5  ll.t*.;  1(^1  .('45 ;  (o  300.  >i-iOii,i.    (<n  2.063 


11.1' 


("I  -345 


45S;    U)  2»i.04. 


4'y. 


Ill  the  preciM 


dint 


j;  c.xaiiiplc  how  niiicii  w.itcr  must 


the  till  hiiie  p.iss, 


wliiMi  hvdraiilic  resislanci'S  are  taken  into  accoaiu,  to  ^ive  the  ilelivcry 


>l  4II. 


/Ins.  1.01; 


ft.  pel  second. 


50.   A  ceiitrifuiial 


-flow  turbine  with  equal   inlet   and  outlet 


ilepthsaiiil  workiiij;  under  liie  iiead  of  Juo  ft.  over  the  iulel  passes  1  cu. 
tl.  of  water  per  second.  The  aiii.;le  ^^  =  13°;  j/i  =  4>  .■  ;  and  the  velocity  at 
outlet  is  radial,  i.e.,  <^  —  90°.  I'iiiii  i.O  tlie  peripheral  speed  ;  i/'i  the  lipaT)>;le 
at  inlet;  (tjthe  lip  allele  at  outlet;  (</')  the  areas  at  inlet  .uul  outlet, 
(r)  the  eflkiencv,  t,ikini4/j  —J\  =  .125. 


,7'  iS';  (,)  iS    40';  {,n  .3623 


.Ins.  (.11  55.215  ft.  per  second;  i.'t  15 
scp  ft..  .0428  >^\.  ft  ;  (- )  .-7''^. 
51.  A  teetiiliiyal  iiiwaid-tlow  turbine  with  an  eliicieiicy  of  .So  per 
cent  and  working  under  tlie  head  of  2tA.)  It.  p.isses  1  cu.  It.  of  wat<  1  p(  1 
second.  The  an^le  y  —  15";  4' 1  —  5>a;  and  //,  =  I'l.  Find  (.o  the 
peri|)heral  sjeed  ;  (/■)  the  lip  aiij;les  at  inlet  and  outlet  ;  ic)  the  iiilel  and 
outli  t  .ireas;    (</)  the  useful  work,  takmn /,  —J\  —  .125. 


.■/;/.,  (.()   55.215   ft.  iM-r  second;  (/>)   157'    iS',  32°  2S' ;  (<  j  .03O23 


A\.     It, 


,0<>  >i\.  It,;  (</)  baSSr,  It. -lbs 


'> 


m 


"ffc 


tS 


ts^ 


ClIAITilR    \-l. 
\'i:kiic  \i.  WATi  !<-wiii:i:i.s. 

1.  Classification  of  Water-wheels.  W,iui-ulnrU  arc 
lari,a'  vrrtical  w  lu-cN  whuh  .ire  iiLulct..  tuiii  <<u  a  liMii/mUal 
axi-i  In-  water  talliiiL;  tri.iii  a  lii-IuT  tu  a  iowcr  lc\cl.  Thcsr 
\'.  luH-U  may  he  iliviiird  inti'  ihrcc  cla-^sc<: 

((/I  /  ihlcrsliot  W'lurls.  in  wliuh  tin-  watci-  is  rciTiwd  lu-ai- 
the  Ixittom  and  acts  l>y  iiiipnl.\<  . 

(/'!  Ih;;t.\t  117/, r/s.  in  win',  li  the  \vater  is  received  a  little 
\)v\n\\  the  axis  ,,1'  rotatnm  and  acts  />,/;•//,'  hy  impulsi  ,tii,/  fnif/v 
i<y  its  wrii^lit. 

\i  <  ('>:;rs//-'/  //'//,•,/,,■.  in  uhicli  tin-  water  is  deliscn-d  nearl\- 
at  tile  tnp  and  acts  ,l/i,llv  hy  ify   -,;i"lit 

2.  Undershot  Wheels.  WheeK  nt  tins  class,  uuh  pi. me 
tl'Mts  (H  l.ii.kets,  are  siniple  in  cdnstiiu  ti^n ,  are  easily  kept  in 
repair,  .md  were  in  mneh  Ljreater  iisr  iMrnurly  than  they  are 
"""  ille\  .Ml'  still  fnund  in  reninte  ihstllv  ts  u  iicre  there  is 
aii  ahnndaiue  .  if  \\  .iter-pnw  er,  ,ind  ,,re  aU,,  empli.\td  tw  udik 
ll'.atniL;  mills,  t,,r  which  purpose  they  .ni  snspcnde<i  m  .m  ..pen 
current  In-  nie.ms  .,("  pik-s  .,r  suit,il)l\-  mo. .red  harj^'cs.  Ih.  \ 
are  ma<le  fn-m  lo  tn  :; ;  ft,  in  diameter,  ,md  the  Unats,  which 
at.  h'.m  2.\  t..  jS  ins.  deep,  are  lixi-d  either  n.  .ini.dK  t. .  the 
periplicr\-  of  the  wheel.  ..r  with  a  siif^lit  sl,,pc  t.,wai.ls  the 
siippl>--shiice.  the  an-lc  hetuceii  tlu-  11. mI  .ind  radius  hciii'^ 
tn.'u  If;''  ti.  30  ,  The  ijeptii  i,f  .1  il,,,it  is  ii.,m  one  tnurfh  t.. 
otu-  lift!)  (.t'the  radius  .md  sli.inl.l  n..t  i)e  less  than  i'i><u\  \2  t.. 
14  ins.     -Il, cy  arc  from   14  i..  16  ins,  .i|i,ut.  .md  -ener.diy  from 

416 


1 


CNDfiRSHOT   IVHF.l-lS. 


1«7 


•(inc  liair  tu   i>iic  thini  of   tlu-   intdl  tlcplli  d'  float  is  acted  u])()n 
li\   tlu-  \v  atcr. 

I. ft    Vv^.   2i<'!  rri)r<.---ciit   a  uliccl  \uth    plane   floats  working 
in  an  open  current. 


Let  :  ,  he  tlie  \eloi.it\    nftlie  t  iimtit. 
Let  l(  lie  the  xelmity  >  >t  the  wheel'--  periphery. 
Let  (J  he  the  ilelner}    ol'  water  in  euhie  t'eet  j)er  second 
The  Neater  impinijes  upon  a  llnat.  i-^  i-educed  to  relati\'e  rest, 
and  1^  carrie<l  aloiv,'   w  itli  the  \eioi.it\    !i .       1  liu> 


ind 


the  impulse  —         (r',  —  //\ 


the  useful  work  i>er  second  —         u{ 


9  'm 


ll.iice 

I  I 

winch  !■>  a  inaxununi  and  (iiual  to       when  ;<  =^     ',',. 


4lS  LMIfKSHOT    U'HFI:!.^    IX   .1   STR.IICHT   K.-K.H. 


t     I 


I* 


Ilu'oix'iically,  tliciolMrc,  tlu'  w  1k-c1  vM'ik-  t>>  tlu-  hc^t 
advantaL;^  w  iu-n  the  xrlmity  nf  it-,  ]  ic:-i])lK:r\  i-  i.nr  lialinr  the 
iiirrcnt  vcli)i.it\-.  I'',\in  then  it--  maMimiiit  'lu-' iii-tiL  crfi-cl  is 
"Illy  5fi  i)cr  cent,  and  m  piactuc  thi->  w  L,'rcati>-  rnhucd  \>y 
IrutiMiial  and  (.thcr  losses,  s,,  that  the  iiscfui  (.■iTcrt  i.ireK- 
cxlcciN  3C)  per  I.  lilt.  riiders|i,,t  w  IktU  w  itli  p'.iiu-  ilnat--  arc 
Cunihrous,  lia\e  little  et't"K  leiicy,  and  --IkiuUI  in.t  he  used  (>n 
falls  ,,f  ninre  tlian   ;  tl. 

.v-am,    let    ./    lie    the    water-area    of  a    tl'Mt.   a'ul    re    he    tli. 
speeillc  \\eiL;lit  ■  if  the  i\ater. 

:,■(]  IS  somewhat  K--  than  re./.",  ,  a-  tliere  will  tu'  an  eseape 
of  w, Iter  on  hotli  --ides  ot  the  lloat. 

I.et    :i'Cj  —  h^'.lr^  .    k   \w\\v^    sonie    eoeftRient    ^<  ii   to   he 
deternniied  1j)    iwperinie!  i         Then 

tlic  Useful  \\oik  per  >eeond  =-  KAw    '  (:     —  it) 


am  I  It-  ni.ixinium  \  ,due  = 


4,C 


Aceordin-   to  Hossut'-  and   I'oneilit  s  expMi'innts  a    nie.m 
valu.'  of/'  i^   _,   and  the  hesf  elleet    is    oijtained  when    k  ~   ~  r 


m 


tlu-   eorresponclin;,;    u-^elui   w  oi  k    h 
4^ 


:4   ■-'•^i'- 


ij; 


and    tile   edi 


3.  Wheels  in  Straight  Race  -  f  .enoralI\- fh<-  w.iter  is  let 
•  111  to  the  wlieel  tliroUL;h  a  ehaniiel  made  for  tile  purpose,  .md 
Josel)  tittin-  the  wheel,  -o  ,is  to  pr(\ent  the  w.iler  ese.ipm:.; 
without  doiiiL^  Work,  I'or  tlu-  re,i-oii.  .d'-o.  tlu  ~p.i.,e  hetween 
the  ends  ol  the  ll.Mis  in  thru-  l.,w.e-t  |)ositions  ,inil  the  eh.mnel 
1-  m.idi.-  .1--  -mall  .i-  1  -  pi  .u  tn.ihle  .iiid  -liould  not  t  \( ced  j  m-. 
JleiKc  /■,  and    therefore   also   the   effkicne\-,  will  he  ineieased. 


IMiF-R.SHOr    Htll-:iLS   IS   A   SIR.-tlCHT   A'.V 


Lh. 


41'^ 


A^'^iinu'  tlu-  cImiiihI  u>  1 


)(■  (.1  a  unit 


"rm  rcctanLiiilar  sci.'t 


t^  •  iia\ 


ion  aiii; 


I    a  l)c(!  lit    s(i   vli"lit 


li.L;lit  a    sl,i|)c  tliat  It  in,i\-  I 


)i-'    rc-'Mrd 


liiri/oiUal   withniit   scilNihl 


f  1.1  Ti  ir 


ii-  w  Ir-1.1  is  usiiall\-  trmii  2a  t 


t(i   48    lli.at-,   litli 


4  to  4.S  tt.  ill  ilianicti 
riir    tli.ats 


20  Ills.   ( 


i-rli,    ii|- 


i.r   railial    ur   iiu1iik.(1. 
aliiiiil  2h  t.i  j;  tmifs  tlir  iloi)tli  .it  tl 


an     \2    tl 


uv^  stream.        [hv   tall  sinmlil  ni.t 
be  radial,   I-'i-.   2  X7 . 


K'  a|i|)rii.n.ii- 


.xcecd   4  It.       J,i.t  the  (1 


oats 


I.ct  //,  Ik   tlK-  lUptli    i,f  tlu'  uatrr  on    tlic  /.-/-.s/y, ,,.;.   si.jc-  of 

tin-  w  luxl. 
I-it  //_,  Ir    the  ilc-l-th  iif  the  uatcr    ,  ,n    th<'  ,/,..-,■//-,;/,■,,/-//  ..idc 

ortlle  uhcrl. 
I-it  ,'    h.   till-  uiihli  iirtJR-  race.        Ihcii 


^^'x'\  =  <3  —  ('/ijt. 

The  niipuh,  -  imimlse  due  to  i  han-e  ofwhuitN 
-f  impulse  <lue  to  ehan---  of  press 


essurc 


"'-1'  ","*/^--//,0 


")+^^^). 


4^0 


i:\-iMnj:. 


A' 
riu,'  tiit.i!  a\  .iil.thlc  w  in 


=  impulse  X  ''  =         "  -''i  —  "J  +  v "• 


H'-iu  r  tlic  cfl'icu'llC)' 


211  IS  II  1^1,         //, 


It 


rill-  -^ctoiul  term  i-^  nc;.;.ili\  c.  '-incc  // .  >  //^  ,  M\i\  the  niaxi- 
niiiin  tlu'i /retic  etticieiuy  ni.iv    hi:  ca--il}'  shown  tn  be         .  v 

Kx.  All  'iniU-rsluil  uiiccl  with  >-' raij^'lit  tlo.its  ai)(i  wci)4iiiii!;  if.ooo 
1'k.  uiHK^  111  a  ri'i  Lingular  rliamirl  with  hcinzoiiial  he'il  and  of  the  saine 
wplth  as  Ihi  vvliccl,  \  ..  4  it.;  th<-  >t'rani  dchvcrs  2S  cu.  ft.  <  f  watiT  per 
s.-i.ind.  and  tiie  clllcicniy  el  the  Wiic  1  is  J.  I-'nul  the  rcl.iti..n  hi-iwciii 
liiC  up-^Iumih  ir'ii  aii<l  ilciwii-strraiii  i«)  vci'nit  ics. 

If  the  "p-strcani  ■eldiity  is  20  ft.  p<T  scchik].  IumI  tlic  d' uvii-^ircatn 
\elocitv.  if  till-  lii.inirtiMS  of  thi' wlicel  and  beanrif^^  are  20  ft.  and  1 
ins.,  respi .  tivi  Iv,  .md  il  tin-  coetficit'iit  of  Inctiun  is  .ooS.  determine  the 
int'cliaiiical  eltcct. 

28  ^  4//ir'i  ^  ^//iii. 


Theref'irc   tlio  •^tli<  umkv 


/h  =  '       and      //,  ^-   '. 

T'l  .'( 

211 


..(■ 


10    + 


7\        I 


7'.' 


If  7'i  =  JO  ft.  jicr  M'CDiid,  tiicii 


.'"1    1     -     1  I  '  .      .    ]    ~       ■ 

■■'■"  3 


/  7    ::o  +  ;(\        1 

\         ^5       "-     I        3 


It  i^['>'indl)\   Lii.il  iii<tt  it  lu->  hcu\  cm  ^.v  aiii !  ')  .n)(]  i>  very  aptiroxi- 
tivilfly  5.'y7  ft.  ]>i-r  second. 


Tlic  total  availahle  power 


(>2i, 


:8  . 


'"937.5  fi.-lhs.  per. 


Therefore    tlu'.tctiial  nicclianic.il  effect   -=     (10937.5) 

,1 
=  3645  .S3  ft  -lbs.  per  sec. 

The  work  ah>'irhcd  hv  hcariiit;  friction        .cxjX   •    15000  >    5,97   x 

-;  1 1.94  1  t.-ll)s.  per  see. 
The  nc    Iclui'iy  111  ft. -lbs.   =  3^'45-83  -  11.94  =  3633. S9. 


I.OSSI  s. 


421 


—  1' 


ur  jiniK'ip.u  I'i'^M---  may  be  cnn-^uliTcii,  vi/ 


UtccI, 


w 


(1)    'Ihc    loss   tif  (\  cubic    tcct  of  tlic  (kLjRT   tliiiil  dtiiuiit- 
liicli  (Id  not  iiiipiiiL;!.'  ujidh  ^,,1111   .it  tin-  liirLiiKist  tlcii-. 


Ac 


inlini/  to  ( icr^liuT, 


0, 


V 


i 


«,  bcini:  tlic   luinibLr   nt"  the  lloat-   minur-cd,  an 


1,  ami  I    b 


accnrtiin 


IS    till'    bdttciiii    (if  tlu     race    is    straii,'ht    '.r    f;: 


wvcii 


abfuptl)   at  tlK   iciucst  pcint  .4  the  uhr- 

(2)    'I  he  l(i-s  of  (J,  cubu   tcct  nf  \\  at  jr  wliich  escape-  bt  t 
tlic  wheel  .iiul  the  tacc-bcittimi. 

Approxiinatclw  the  ])ia\-  at  the  bottom  may  be  said  t.  v.iry 
from  a  minimum,  .v,  -  />'('.  when  a  tloat  J />'  is  in  it-  '.oucst 
position,   bii;-  -v"^.  t"  <i  maximuni,  />',(,  ,  —  L  1>  —  /.\(     ,  \\hen 


two  floats  .;/)',  ,  ,-/,/)'.  are  e(|ui(listant  fron.  the  lowest  position. 
I'l^     -3^-        I  bus  the  me.m  1  Icarance 


'(/.'C"-f  /.',(■,)=    '(.V, 


CD) 


I  1  {/>',/^r 

-A2S,  -t-  I'/y)  ~  s,  -f      ------    ,  nearly, 


4 
if. 


;♦'. 


r,  biiTiL'  tlic  wheel's  r.uiius 


4?; 


LOSM.S. 


itf 


l?llt 


-  T/ 


^^  ili--t,llli,-r    hctwccn    tun  i  nii,otliti\  r   n<KltS 

—  2  .  /■',/'.   M-iy  !!!Mrl\  , 
>.'  hi-iii-  the  t<.t,ii  niimlur  ..ril.Mt-        FIciicc 


/'■./'  - 


and  tiicict".u<-  tin-  iiK  ,in  <  Icir.iiac  =  v    -i-    '  ""'' 

■  '  ^   .»    "■  ■ 

.\t;am,   tlu-  (liil,iMur  ..f  l„,ul  ..,;    ilu'  ii|.-tr<Mm  and  dowri- 
'-trcaiii  siiK-s 


.  /,  -  /.  =  /,,l^' ; «). 


nnd   tlu-   v.'l.Kity  of  disc  hai-c.    r^.    ihr^nvj^U   tlic    clearance 

J,'l\rti   l.y  tile   i't|liatii)n 


</   = 


W  / 


Il.-ur 


/  I 


Q:  -  ^vs  4 


4    //-'  /'  '■• 


Intr..di,.in-    ;  as  a  i-.H'tVh  unt  (,t  hvdiaulu    rcsistaiuc. 

Il  tlu-  diptli  III   tli.-    -trcam  i.  ili,-   „,;nr  ..ii  l)..r|,  ,,,1,.,  ,,ft|,e 
wlici-l.  i.f..  if/-,  -  //..  tlu  II 

•''  =  '■•• 
(.0   The  loss  of  (_)^  cubii  fiif  ..t  water  whic  1,  iscapo^  between 

tlu-  \\  Iiic'  .mil  the  raee-sidcs 

I  it  >,  l)e  tlie  ile.irance  on  i-ai  h   ;i,|.-        j  h,ii 
(J,  =  .;  X  2//,Sy:r  -  J  4''/,^,. v. 
7  Iiciii^'  a  1  i.i-nieieiit  of  h     Iraulic  rei+intancc. 


MICH  I  Ml     11     f  /  IfCT. 


423 


'.4]    I'in.iIIy.   it'  II    11)N.   i-~  tin-  wriL^ltt    (Hi   tlu    w  licil -j.  nnii.il 
till'  lo-^-^  due  ti '  iiiurii.il  ("1  ii  tion 


=  ////■     //, 


//    hrin^'   the    imirii.il    i  ( Kl'ticiiiU    ot    Irittiuii,   and    ,/   tlu     j.  lurtial 
r.idiiis. 

Ai  t:i<il  Pi  Ir.  t  ry . — TIiii^  the  avtii.il  dcli\(.M-\    of  tlu'  w  luc!  ill 
|iM)t-l)!)Ululs 

\li\V,  —  //)        Itlll,        ll,\  i    \         ^  i  /) 


wM  f 


\ 


l\tiiiii)i{s.  Mil'-,!.'  \\lucN  arr  nin-t  di'fri;i\c  in  pniu  i|)li  , 
its  tllL'V'  utili/i:  i)td\  al)Miit  .Mil  thud  nt  thr  tntal  available 
I'licrj;)'.  I  lu\  ma  hr  made  lu  work  to  soiiie\\!iat  intter 
ad\'aiitai,'e  liy  iiurndium;.;  tile  ImIIiivjuo  inddirieati'in-- : 

'.'  ill'-  ~iiji[)!y  ma\  he  -c  reL,Milati  d  hy  mean>  ota  sluiee- 
Imard  that  t!ie  ,iiean  tlikkne'-s  n|'  the  im|iin;,Mii^f  stream  is 
alxail  ii  cir  S  iim.  If  the  thiekne->^  i-  too  ^mall.  the  relative 
Id---.  o|  \vater  aloii;^^  the  i  lianiiei  will  lu-  \r\v  L;reat.  It  the 
lllicklU's-,  Is  too  i;ieal,  the  tloat -,  a--  tlie\'  emer',;e,  will  ha\e  to 
raise  a  heavy  weight  o|  water.  i  he  >liiu  e-hoan  I  is  iiu  lined 
.It  an  ailj^le  of  \0  1040"  to  the  veitieal,  s,,  that  the  sliiiee- 
Mpfiiin^  ina>-  l)e  as  near  the  w  lu-el  as  possil)],',  thn-:  diminislnn'r 
tlir  lossdfliead  clue  to  ihaiiiu'l  tVietioti.  and  is  r..iini!rd  at  tlu: 
1  lot  loin  til  pi  e\  ent  a  i  onti  ai  tion  ot  the  issuing  fluid  \e"li'Ct- 
uil;  Irutional  losses,  etc.. 

"'1'  w'^  \  loss     of      eiRTf^y 

iliic  to  shock 


tlie  usetiii  eflei  t  ,  1 


=.:..0(//,    --"")-.  ^'<^(^'.-'')* 


H  l-ein^;  tin-  dillereiice  of  level  between  the  point  at  uliicli  the 
water  enters  tlic  wheel  uiiil  tlic  surface  ..f  tlu    u.itn  ui  tlic  tail- 


1 


4»4 


M/<Jl.tM(.  -U    I  ff/cr. 


»        ri. 


^ 


race,  i  (.■,,  tlic  fall.  H  i--  ii-^n.illy  \ ci},-  --ni.ill  aiiij  iii.iv  lie  in'L;a- 
tivr. 

It  llic  \ani--  ail-  iiiclnuMl.  tlir  i(--istanc  c  \i>  i  iiiciLjtMicc  i'; 
not  so  L,Tcat.  aiiil  the  !i  ii  tic'ii.il  luil  n  ■-i-^taiu  i  hetwi  i-ii  tlic 
sluice  aiul  tloat  i'^  practually  rcihucd  to  ////.  With  a  straiL;ht 
bed  a:iil  small  sIcp|K-  'i  m  lu,  the  !iiiiiinuiin  coiucniiiit  diaiiuter 
of  u  ileel  is  alioiit    14  It 

(/')  I  111'  1"  '\  ••\  the  I  hanin  I  fur  a  ilistance  at  least  e(|ual  to 
tile  iiiter\.il  l)etueen  t\Mi  cniiseciiti\  e  \aiU";  ma\-  he  ciii\t(l  to 
tile  form  of  a  circular  arc  concentric  with  thi'  wheel.  \Mtli  the 
\lew  of  ]ili  \  elltmi;  the  (si.iin-  <  it  tlu  \\  at(  1  until  It  has  exirtei! 
its  lull  clliH  t  upon  lh(  wlucl,  \\  lit  11  the  lird  |s  tiir\e(l,  the 
miuimmii  iiin\(nirnt  .lialiieter  oi  wilcel  |s  al)out    U)  It, 

All  uiuli  ishot  wlucl  with  a  curb  is  in  realitv  a  low  hre.ist 
wheii,  and  its  theory  is  the  same. 

((■)  1  he  (lo\\  n-stuMiii  I  liamii  !  iiiav  lie  (lei]n-neil  sntliat  the 
vi'Iocity  <il  the  water  .is  it  tlows  awa\'  Ihh  nines  ^  ,- ,  1  In- 
hnf'uls,  liin  !■■  ^iissttr,  is  then  positive,  \\hi>.li  inciiasts  the 
iiselul   w  n!  K  .iiu!  therefore  also  the  efficienc)'. 

Ui)  The  down-stream  chaniul  may  lie  wiihiuil  .mil  a  s|i-ht 
counter-inclination  j;i\en  to  tlu  lied  W  h.it  is  known  as  ;i 
i7rt//<////j,'^-ii'(/;<  is  then  proihaed.  in  \utue  ot  vvlmh  tlu  re  is  ;i 
jiiidden  rise  of  surfacc-U-vel  on  tlu-  dow  i\-strtam  sjcK'  ,d)o\c 
tlt.it  I  a  ihi'  ii|i-stri'am  ^m.\c  Ihi-.dlous  of  thewh.il  liiin^ 
lowered  I))'  an  anuaint  e(|ual  to  the  difference  ol  level  hitwecti 
the  surfaces  of  the  staiulin^-wa\i'  aiul  of  the  water-layer  as  it 
leaves  the  wliecl.  thus  givinj,'  a  corresponding  i;ain  <if  lie.id 

(<■)  'IIjc  introduction  of  a  sudden  fill  has  lni-n  .uKoeated 
in  order  to  frte  the  wheel  from  hack-u .iter,  hut  it  must  he 
borne  in  mind  th.it  ,ill  sucii  falls  diminish  the  available  he.id. 

4.  Poncelet  Wheel  (l-'ij^s.  2y),  340.  Thus  undershot 
•wheels  uitli  ll.it  hucki'ts  li.uc  ,1  sni.dl  efl'u  lency  because  of 
the  loss  of  enerj;y  in  shock  .it  enti.mce  ami  because  of  tlie 
loss  of  ener^'y  carrieii  awa\  by  tlu  water  on  ieavinj,'  the 
Vfhecl.       These    losses    ha\t    been    considerably    modified    in 


,4 


rOMJI.I  /    li'l'hFL 


4»5 


■let';  wliri'l,  whirli  is  iiitiii  tlu   hr^l  innt^r  tn  lulMpt  ulu 


the    laii    <liit-~    imt    rxcccd    d 


h    tt. 


,llRl     U 


Inch,    ;n    Its  (1 


CSUM,    IS 


V 


Fig.  210, 


Kiii    2411. 

(;ii\(ilU(l  li\-  !\\M    ]iiiini|iK'      tll.lt     -lnu'ltl  ;.;i>\,|-ii  cvci)'  pcrlrit 
\\ .itc  1  -nii)t(jr,  viz.  : 

I       I /iiif  till    loss  o/imrj^y  i»  s/iihL'  iit  iHtrana'  sfioii/u  /»<• 
u  iitininniiii. 


42b 


r  ox  a: /.FT  lynhi.i. 


*  f 


{2)  77/,?.'  ///,■  Vi-Ui/y  of  til,  i.aUr  „s  it  loavrs  the  ivhcd 
should  !h   <i  iiiiiiiniinii. 

II"-  x.iius  ai-f  cui\i'(i  ,111,1  arc  (.ompriscd  i„-t\vccii  two 
nnun^,  ,it  ,1  -lii^htly  -rcitrr  clistaiuc  apait  ihaii  the  vanr- 
\Mclth;  tiu'  inner  rn,]s  ,,f  tlir  vaiu -,  arr  radial,  and  tin-  ualcr 
acts  in  ncarl\-  the  -aiiu    nianiUT  a-,  in  an  impulse  tnrhinr 

A  I'om.'ict  ulu-cl  Mt  ifMin  lo  ti.  I  :;  It.  m  diaimttr  has  ^6 
floats,  ^^l,iK■  f,,,-  ul,,,,!-  of  Inun  2o  to  j ',  It.  in  di  inu-trr  tlu' 
numluT  of  floats  is  alx.ut  4S.  1],..  ,^hc^■U  aiv  usnall>  lioni  10 
to  20  ft.  in  (lianictiT  and  have  from  ;;_-  to  4S  tl,,,,ts  ul,icli  may 
be  of  platt-iron  or  wood. 

/■list,  .\ssuinr  that  tin-  outer  .nd  of  .1  v.iiu-  i,  t.in-cntial 
to  the  uhrci's  jHripIu-iA.  th.it  the  nnpniLjin-  l,i  -r  is  inhnitrly 
tlun,  and  that  it  -trikc.  a  ll.i.it  t,in,;;i;nti,dl>-. 

I.ct  -(/I  I'i.i;.    J41      1).    .1  llo.-it,  ,ind  ,^/  th,    t.m-rnt  .it  ,; 

I  hi'    M-loi  lt\     ..|     till     \\A\r[      r<  l.itivcly 

y     to  till   ilo.it       .-^  —  /,. 

I  he  uatir.   in  \  irtiir  of  ihi,  \,-|oc'itv, 
asci'nd-     on     the    lnu  Let     t,,     .^     hri;dit 

(v.  —  tif 
^''  '-'-       \,r       .  tluiHalMMik  ..nd  haves 

tiu-  lloat  uith    th.    r,/„ttr,    \rl,Hit\-  ,_  -u  .in<l   uitli    ,in  „h.u>/i,/,- 
velocity  :■,  -  2/1.       Ihi.  .disohitr  m1i,uI\'  ,.  nd  v\h,-n  the  speed 

ofthi-    uheel     i.    suU.    that   I,   .-:   i.-,  .    ,ind     ,lu.     throrrtu.il     hrl     ht 

..I     a    ll.Mt    IS    /.,;  -=        ^1^.        |1„.    ,,,,.d     .,\,,,l,.i.h-    lu-.id    I.    thus 

4  -.^ 
cljaUKcd  into  useful  vv.,rk,  .md  -he  <  fdv  i.mu  y  is  „„//,■,  ,,r  perfect. 
I  akin;;  A'   a-   the    mean    i.nliiis  of  tin    iroun  aid  //„,  ,k  the 
correspoMdin-    h  lear    v.do..-it>-.    th,     me.ni    .  rnti  ifiu;al    I.,,,-    ,„, 

cat!)  unit  of  tl.n,!  mass  is  "'^^  ,,n,l  aets  very  nearl\  ni  the  direc- 
tion ofjrravity,  sothat  the  hdKlit  />,/  of.,  il,,,,i  m.iy  !„  ,|,p,oxi- 
m.itely  express)  .1  in  t'u'  form 

/■■• 


► *.  a_ 


Kii..  341. 


/'/ 


^'■^+';?Y 


,4 


I'OM.-fl/:/     HUl.l.l,. 


4-' 7 


rbcin-  tlu-  vcl,irit\    \Mth  uin\li    tli.    v\,itcr  cnmiiicnci-^  to  n^c 
'in  tlu-  t1ii.it. 

i'r.Ktu.dly,  li.,urv.T,  tiir  ilwat  is  ;/,,/  t,m-ciiti,il  t..  the 
ncriplKTV  .It  ,?,  .IS  till  u.itn  ^nvM  not  tlun  ,  iitcr  th.  uim  !. 
\Uo.  t\u-  iinpin.i,nii!^'  u.itcr  is  ,,f  scnsihK-  tliiLkiir-s,  strikes  tlic 
;i<;rililicr\-  .it  smnc  .•ii)prcci;ihlc  .iii-lr,  ,uul    i;i  vWnv^   .md  lallin^^r 

"II   the    rlo.lts    |.,scs  ciUT-y   in    ,hncks,    edchcs,    ,ti 

I  .I'l  thi  w Mtcr  inipiiiL;!-  in 
the  du'ection  ,?, ,  |■'i^,  :'4j, 
.md  t.ike  r^        r-, 

I  ;ike  (t,f  in  th<  <hieetiMU 
"f  .uk!  e,|ii,d  t'l  ,7,  the 
\el<iv.it\-  ol  th(  wheid's  pe- 
iipher\ 

<'>inipK  li     the    ]).inllehi- 

j,Matn  A/. 

I  hen  ,,f       ,,,'■  —   \"\<  the 
\eloiUy    .'.I    the    w.iter    re!a- 
<     tl\tl_\    tM   the   (1ii.lt. 

p 


Kk..  243. 

Ih.it  tlieic  in;i\-  lu  !H)  shnek  .it  (iui,ini.e,  ah  must  be  a 
;.in)^ellt  !■>  tile  \;Hle  .it  ,}. 

A^'ain.  tile  u.iter  leaves  ih,  x.nie  n,  the  diieai-n  of  /■,, 
produccti.  .ind  uith  a  rel.itive  \<  locity  ,..   =  ah  =  / '. 

Complete  the  par.illelu^'iam  iL  .  I  h,  n  ./.i.- (  =  r  )  is  tlie 
.//^,Vf'////<- velocity  of"  the  w.tter  leaviiij^  llu    u  heel, 

ICvidently  alf^  is  a  stiai^dit  line 

Let  (h.    .in;^le  ,,ui  =  y.  aial  the  allele  bad  =  n       „. 

I'rom  tlie  triati^Mc  aih  , 


1> 


r«  =:•,"+,<-;,,/,  c 


OS     I 


I') 


428 


rcsciti  '   H'niiiL. 


¥m 


7'*  —     r-  4    li-   —   2  I'll  C< 


S!I)    y 


J'roin  the  tn.m^lf  ,/,/^', 

~''i    =    ''■'  -r  "■  -\-  iVk  cos  rt-. 

hv  r.|ii.itinns  (^  1 ).  (2),  and  14  \. 


13) 
(4^ 


J/'V   Cn^    ,►  =    ,y  ---    J -J  _   i/i 


1  IriiIhii;  tile  ii^rtul   work   pir  ^iMiin! 

=       ^     2//17'l   COS    I'  —   /n. 


ittir^  COS  1'  —  /<). 


•     .     (5^ 


This     is    a     m.iviniurM    .iin!     cinal     to     "^''    ^"■''    )' 

.'•,   cos    )' 

"=  ,        .   •iiiil    'lie    iiiaxiniiim    dVu  uiiv  \    iscs'j-.       | 

too,  iIk  aiii^ii'  ,/,//'   -^  90  ,  ami.   I.y  I1-.   .'4^, 


\\  hen 
iciicc. 


tan  I  7         II 


lit/  ad 


-  tan  ;-.         .       .      (6) 


Also, 


V       ah 

,    =      ,  =  sec  (n  -  a  I. 


■     •     .     .     (;■ 

I  h.    I  fCu  irtu  >■  Is  jiiTfut  il"  ;•  i^  nil.  .mil  tiler. -li  ;rc  tx     .   iSii 
I'l.u  tu  .ill>   this  I-,   .1,1    iiii|).isv,|,l,.   \,il,|c,   hill  the   preccdmej  cal- 
culations   iiulidt.     that    r    s|i,„il.l    not    l.e    t..,,    lari;<-    Misn.i!I\- 
<  30"),   .111.1   that   the    sp.x-.l  ol  till'  uheel   s|i,,ul.l   he  ,1   littk    less 
th.ui  one  halt  •  .1  the  \el..iaty  of  tii.'  mil. .win;;  stri.im. 

1  .ike   r    -    I  5     .Is  .1  nie.ip   \.ilu.  111.  1) 


('■^ 4-  •I'l'i  th.    I  If 


CKiicy  ^  .yy3. 


roNci-iHT  ii-Hf:/:i.. 


429 


T  iic   l)c->t    practio-    iiidii  ,itc^    tl 


K'    relation     \  1 


"'V,        It 


must  l)c  bdi-nc  in  iiiind  tliat  tin-  tlicur)-  ap,)l 


P,)Ik'^   to  on 


c    i  Iciiii.n- 


tar>-    l.iycr   only,    say    tlic   iii<  an    ia>i-i.  ami    tli.t    all    the 


a\crs  i-ntrr  the  uh 


eel  at   aiu 


iiiUTiiiL;  liom   I  ;  ',  th 


rise  to  •■  lo-,M-s  ntCn,  i<'\   in  ^iiock.   "       11 


ll       OS-NlS    of 


!rii  tionai   r<  si-,taiK'c,  nlily    motion,  rti    ,    in 
!ia\c'  .lUo  hern  (li>r''-,u(l(  .1.       I.mi'eiitial  ent 


us   'MMllL 


ciicri,')-  111 


h\c  in  nr.ietiee   ,in<i  the   (  ft 


tile    \  ane-|),is-,at;c-^ 
laiu  c  IS  n.  t  |)()ssi- 


u'H-iu\   lto(■■^ 


not  eveeed   ,6:;  Im,-  f.ill 


iip  to  4  tt. 


IS  . Oi)  lur  t,ii!  -  ot'  iVoiii  4  ti 


5    5 


ft.,  an 


111     ll  (ini 


to  .5(1  for  f.ilN  ,,t  •roni   ^   ;  to  «.  ;  it         !  h, 


ater  etticittic\-  of 


the  I'oneelet  wheel,  a-  eoinp.ueil  v.  ith  wheel-  ha\in'_'  l1at 
l)uckct>,  \er\  i.leail\  show-,  tin-  impoitanee  nl  brinL;in"-  tlie 
water  on  t<i  the  wheel  in  -ueh  a  inannei  ,i-,  i.,  ,i\(,i(l  ln.~,  ,,\ 
cner;;>   in  ^Inak  .iml  in    th.-  |)roiliiction  ,>i  edilie-         I  he  1, 


of  \^atei ,  ll 


w  ith  a  \  eli.eit\-  \  er\-  p     .r 


i\  er-. 
uu  e  aiul 
i>  e(in.ii  to  tiiat  due  to  the  total  head. 


owiiiL;    to    tlie  wjieel  undei     an    .idjii-tahle  si 


ma\    lu-    all    mad 


(       i  o     I 


ntei    ,it    an-i'-,    ap])ro\mi,i|rl 


15  ',  and  the  eoi  le-pMndm-   io^-e-,  in    -,hoek  redund 
nnim  hy  toimin;^  the  miowi  ,1-^  lollow-.: 

1  he  fu-t  i)ait  ol  the  enui -e  /(/,  l-'iy.  244, 


>     eijM.il    to 
to  ,1  inini- 


is  i.urved  i'     ,ueii 


? 


» 


tf 


a  manm  r  tli.it  the  nonnal  /■-/;  ,it  .my  ijoint  /■  makes  ,m  aii^'lc 
ot  Is  with  the  1,1dm.  , ,/  I  !„•  water  moves  sensibly  parallel 
to  the  bott,,ni  /■(,,  and  tlieretore    in  a  diieetion    ,it  ri^;lit  .inides 


'if--- 


.k 


430 


roxciiri   a  Hi-rL. 


hi;' 


V:  \ 


t<.  /r.  !  Inu  .  ,it  ./  \hv  <l:rcai<Mi  .,f  .imtinn  m.ikcs  an  ,,:i-lc  uf 
13-  uith  tlu-  t.ui-riit  t..  tlK-  V,  IktI-.  priiplicr).  if,-;  ,.  ,h-awn 
iHilKiulicuLir  to  //,  ilu-n  ,v  ^.  ,.,/  s,r,   ,  _-     ^,- ;,  cnstant. 

lluis  the  n..n„,d    /•,//   tnialics   ,,t  ;   ;,  nrdc   o.no-ntri.    u  iti. 
tlif  \\lu-<I  ,111(1  Mt  ,1  lilt, 111!  i(.ii-,t,im  ili.iiiictn-. 

llH'  initial  ]u,mt  /•  .,t   the    i.r..r,K    /  (,  ,.   tlir    |H,mt  ,n  \v  Inch 

tlu'  tall-CIlt    tn    tllW    Uivic,    pa^MII-     thlMll,;l|    tlu'    upper    cd-c    of 
tllc  Mima'-opciiiM.;,   ciil-  tiic  bed  ..itllc  Mipply-cliamul. 

I.ct    ./  be    tlu-    drplh    ,,t    tlu-    ,/,,-,v/    nr   a/',-,v^,//;/-,    j.,..,    tlx; 
ii'M-nial    (li.,taiu,     hrtw.Hii    the   .mter    and    iiinci 
pnijihcrk  ■-  ..f  the   w  heel. 
Let    /'    lje    the    uidth     and    /    the    thickness    of  the     ^luvt    of 

\^  atei   eiiieniiL;  the  \\  h>-el 
Ihen,  di>re-ard,iiL;  the  tliiU ne-  ..fthe  tlnat..   the  eapaeil>- 
of  the    portion    ..fthe    uheel    pa-in-    in    |,-,,nt    nfthe    entering; 
Mieain    per    .eemid     i,    appr.  .xii.iatel>-   /•„;,,.       iVaetieall)-,    the 
uh<.|e  ,,|  tin-,  -pae.    .aiui-.t  In-  .Keupi\d  by  the  water  .uid 

ll'''l^H,„     7U       (   '     —     /'/," 

»l  bein;;  a  t..etVKi.  lit   \  .ir\  ill;.;   Ironi    '.  tn   -.  . 

■I'iiu--  /,  the  thiekne-s  ,,l   the  Mreain,  111,/"'- 


=  »/i/ 


A'    // 


'% 


N 


If  the  efrieieiu\- 


I--  a  nia\iniiim.   ,■    id 


■^  r        211,  and  then 


,/       e,, 


)'■ 


^■'"'    "'''^    ""■     "'•'•"I    ^^■"'•i-    I.iver    at    the    p,,int    ,.f 


cturaiii  e 

//bcini.;  the  availabl.    fall         iletui 


■.-'V-'-''("--')' 


|;4 


/'()\a:ihi  li-in.i.i.. 


43» 


an    ,iVLTa|.;c    \aliu-    of 
(iia-liof,   //    _    If)/, 


Jcuil;    .9,    aiu!    It,    a^    ai.i.Mr 


illllL'     lo 


^V 


_'-// 


Mc'iin  make-  tlu-  radui-.    /,     ,,ltlic  wlu'c!  Irnni  /,-.v>  to  //ir,r 
tiiiK's  tlu;  .Kpih     ,/)  nt   the  uoun,  and  l'..iRcKt   o  M,->i.lL-i-.  lliat 

tlus  .kptli  -hi.uKl  l)c  al)Mut         aM<l  11, ,t  !r<-   than    ^^       In  ,,nli.r 

^  4 

miKxd,   U,    pnv.ilt    tlu-  uatcr    truni    lisiiiL;  ca  n    tlu-    Inp   ,,t"  the 

// 
tl.uts,  ,/  slu.nl.l    1h-    I,-,, ,11     _    to  "/■/.  ami  tlu-rc-|.,ri-  /-,  Ironi  //to 

j//.  tlu    latter   Ik  iiu;  dlUn  aduptcd  m  piaau-c-. 

N)c-    ana    d    tlu-    dnuT-c .pcinn-   n-ualh    \aru--.   tiMiu    1    J,/--, 
to    I     :;/'/. 

'Iiu-  in--id(    width  nt  tlu-  u  lu-rl   i-,  ahc.ut   , /• -i^  \)  ft. 

ll     \     1-^    tlu     aii-ir    -nl,trndi;d    at  the   ecntrc(',,t    the  w  hcfl 
l'>      till      w  ,ilc-i  -an      lu'tw  t-i  n      the 
l'<'int     ot     ciitianic    ,;     and     the  j 

Idue'-I  point    (  ,    l-i-.   J4 ;.  ol   the  i 

wheel,  and    n   ,/,/    is  ,l,-,iu  n    hori-  i 

/ontally.  thin  ./,/  is  a])])rn.\i- 
inatel\-  tlu  lu-i;;ht  of  the  lloat, 
and  the  theoretie  de|)th  ,/  nf  the 
emu  II  is  .ri\,  n  1)\- 


,/  ^  /ir         .;,/    -f  C,/ 

-     ,      -   ,,  ■>.  +  'M'  -cos  A) 
-t+-A") 


? 


In  piaetue  it  is  \\^.\\.\\  to  ineiease  this  deptli  1>\-  /,  tlu-  llnck- 
ru  ss  ,,f  till    iinpniLMn:;  watei  -  ia\-er,  aiul  therefore 


i^ 


■> 
■<^ 

Ik.    -if 

It  I 


43^  F.FIIcmsCY   Ol-   rONClUMT   IVUHEL. 

.         2         I'' 

"  ^    A' 

I  lie  buckets  arc  u-;uall\  placed  abdut  I  ft.  .ipait.  measured 
a!. Ill-  the  circunit'erciKc.  Init  the  nuiiihrr  ol  the  luickets  is  -ot 
a  matter  ut  -reat  imp.  n  tancc  Then-  ,ire  ^emTall\-  3()  t)uckets 
ill  uht'els  ,,r  1.  tM  14  It.  (hanict.T,  ami  4S  buckets  m  wheels 
ol  JO  to  -\i   It.   diameter 

It  ma\- i)e  a--,ume,l  that  the  uater-arc  i-  i.|uall>-  liixided  b\- 
tile  ln\\e-t  point   (  ■  i>l  the  u  heek   sothat 

the  len-th  of  the  uat.r-arc         2\/,         2ii  J\ 

■/beiiiL;  the   time  ot   the  ascent  or   descent  of  the  water  in  the 

l)ucket. 

In  the  middU'  position,  the  ujiper  end  of  the  bucket  sl.oiiid 

l)e  \erticak  ami  if  the  lloat  is  m 
the  lorni  ot  a  circiikir  arc,  its  radius 
'■  =  '^  sec  (t  —  (t  I,  11  l)eini;  the 
an-Ie  between  the  liucket's  hp  and 
the  uh<(.Ts  periiih<-ry 


O 


I'k;.  24(). 


I  he  time  ot  ascent  or  liescent  is 
also  ^i\eii  b\- 

7-  _  '■)'''  +  ^'"  '/•     /       '-'  ' 


\\  liere  sm 


♦  cos  (  n  —   (1 ). 


5.   Efficiency  corresponding  to   a   Minimum    Velocity  of 
Discharge  (,  ,1..     Irom  ku;.  2^2, 

sin  ,-        ,,^(=4^^)^  ^(rg 
sm  ,?,',/    '  (7(/  u    ■ 

lleiUH'   for   an\-  L;i\en    \  ahus  of  u  and  ;',  r.,  is  a   inininnmi 
ulien  sin  <?f'(/ is  Ljreatesi,  that  is,  when   aod  ~  <jo',  or   ct_i;  i-  at 


h:x.-iMri.F.. 


433 


ris^ht  aiiLjlc-^   to  (i,\      Then   ,iU(,  (ui  —  tu-  —  ah,  or  u  —    V,  and 


.•/(   iM'-ect^  tlu   ,ui: 


rk-  ".^r/.       l- 


;;/■  COS  1        and 


^.u  sin 


I 


1  lie  Li^ctul  work 

w    7.^  _  ■•:■     ir 


'   =         2//'  COS  2y  = 

2  ,1,'-  .<'•     2 


/r  ;,-'  cos  2) 


^^'^     2    COS'  y 


riic  t'  ita'  .u'.ulablc  work 


//■ 


T  lien  fore  tlic  cffu  ieiu\',  ii,  = 


and  the  H.l'.  ol  llic  uiicci 


cos  2y 
cos^  y ' 
(>2hQH 


I'.NlicriciKc  indicates  that  t'  -  uM-t  ta\(.ral)!i'  \aliic  for  le 
he-  l)etueen  .5,-',  and  .O,"',  ,  and  tiial  th>-  a\  rai^e  value  ot  the 
el'tu  K'nt  \    IS  all! 'lit  Oo  per  cent, 

Alth..u-h.  iindci  norma!  condition^  .  it  working,  tlie  effi- 
I  iency  ot  a  1'.  miilet  win  el  i>  ;i  little  less  th.m  that  of  the  he-t 
turl)ines.  tile  a(Uai!lai;e  i-  with  the  former  when  workiiiL,'  with 
.1  rediKed  supply 

i'.\.   '[    ■  ()esii;n  .1  I'.  .iu<-l.-t  wiK'cl  for  a  l.ill  •>(  41  ft,.,    li  ,1  u-.itcr-^^p(>ly 
of   ^4  I  II     ft.    per  second.  t:ikiiisj,  a,s  a  lirst  .ipjiro.Mm.iliin,  y'  —  \°  ==  20*. 
.Ui  in  Vthhity  (r'l)  .it  fituiit  of  aiimt'.toit : 


V,  =  .9^'  3;  .  4i  .        -  15.03:7  ft    per  see. 


/^■-/  .perj  ol  pntphnv 


u  =  ^v,  "  i>g  30'  =  7.0631.'!  ft    pet  sec. 


t.m  IT  —  ,(|  -   I  tan  20°  =  .728. 
and  T  —  i»  =  36*  3',     or     i»  —  143°  57'. 

[  'uluf  of  1/'  ,• 


and  (/" 


sill   '     -    ♦''cos    36'    3'    :=    .89917, 

:8'.0   =  iiS'.i. 


434 


r..\.4MriJ:. 


r      i-\.>A 


i\  :'-\ 


l:i 


If  ■,  ■  i 

mi  ■ 


■»  ! . 


KiUlive  :\-!o,  it)  (I'  ^  .1/  .i.i'm/  y  ^/oii  : 

r  —  ,i  sec  36"  i    --.:  S.73()i   It.  [.(.T  ^or. 
I'li^ur  oj  r,.      T.ikiiii;.  ,i>  a  lii^t  .ij.iMu.xmi.ition. 

J\  _  /■,    ..   ;,/,   /,,  __  /,,   ,,ih1   .1     —  ::o  ,  I  lien 

/•'    =   li  MC     T,(l'    3'    ._    ;  ,     X     .4123,    ;,ll(i 

I  :;.S.  I 


T    „    '  ]  7.0(>-!l,S  


i.So 


wludi  i^ivcs  >-,  ^  7.443  tt.,  or.  s.iv.  7J  ft. 

yv////  (,/)  of  .,,r,',i.      Takin-..,-,  ;,„  ..ppr.iMin.ita  iii.  u,^u  and  A'  =  >-.  . 
2        (.S.  73011' 


J  -^ 


3  (7.061181 


7J(  I  —  c>iv  JO'  )  +  / 


.J2() 


—    '-JTii    +   /   —     1.''^   It..  ■'Ul)[lij--C, 

.1/(7/  f  i,'>r,\  t  I  .i.inis  of  Jl  <.it : 

'■'  =  1.8  sec  s*."  3-  _   ...„-_.,,  It. 

l\i,'iii  ~  of  U  ,iiu/  I/,  : 

I 
'''=/•  5  -   ,('-!^ I    -  '^1.6  ft. 

A'  (  .6 

«,  =      u  =  ;— 7.0O31S  ^  6.2t56  ft    i.ft  v,r. 

-l/orc  ,orr,ct  -o.iiiir  of  \  : 

i  J.S.  I 
';"■  -5-  +  Sin  i:.S'  6'        / 

■*"   ^        6.6 

•I  -^  .2iA'4r9. 
.t°        17M. 
I'iiii  L'ie-~^  {/)  (if  't>f.ii>i  : 

1   i.«  6.6 

/  -    ,     ,    CMS  JO'  .  ^-^  =  .37::  ft. 

Jf '/,////  (/o  of  -(cheei : 

-^^'^--^-^^ ----=., ..ft 
.?72   X    r|;.03:<(        ♦     '    '■ 

'finie  (  /"i  of  ,tM,,il  or  Jos,  fill  of  uolri  on  fioal : 

Au>iib,i  ofjiouts  (.\).      ll  ^l,,|,,■rl   I   ft.  .!|iart. 

A'  =T  jT  .  7.1  z^.  471.  ,,r.  vay,  48. 


.1 


li 


li'ii:'^ 


lOk.M    Ol-    HL:ckl.T. 


4J5 


ritdit elhal  iii.i.x 


imum  /•:• 


(,2!. 


24l7.o(i-,i.Sr    =  4(79.0  It. -lbs. 


(<er  sec. 


7W,,/ 


:i:\u/,i/'/,-  power 


(>rsi'  it.-ih-. 


yfu 


li  It,  y        := 


4<)7g  u 


-  =   .<-.93. 


50 


6.   Form  of  Bucket.     'I'lu-  fnmi  of  tlio  bucket  is  arbitr 


and  nia\-  W-  .is-unuil   tn  1 


n   a   circiil.ir   art 


li 


l>racticc   there 


are  \-,iri(ui-  uutlKKk  i>rti\uiiiL;   it-  t(iriiL 

MiTlloh  I  iIm-.  J.;;  I.      'I'lie  taii-^aMit  r/w  to  tlie  biieket  at 
(I  iiiake-^  a  -i\en  an-le  ,t  uitli  t!ie  taiij^ent 
at  ,t  to  the  wheel-  miter  iien])lier\-.        Ihe  i' 

iMihii-  /'/  is  alsn  ,1  tangent  to  the  hiiekel 
at  /.  It'tlie  ,iiiL;le  ,/,-/  is  klloun.  the  [xi-i- 
tion  ol  /  wii  the  inner  periphery  i-  at  <pnee 
fixed,  and  the  form  of  the  bucket  can  be 
easily  traced.  U 

I.et   tiu'   ani;Ie   ,u'f--    1.       |(iin  ,//   a:ul 
Itt  the    tani;ents    to    the  but  ket    at    ,;    and  Kir..  247. 

/  niei't  HI  ;//.        I'lien 

thi   an;^de  luiiii        a  —  <)()', 

''"I'l    -    1X0    —  tuiiii  ~  aoiii  —  270°  —  a  —  .r. 
wA;  —  tile  an.L,de  W(//  -    Ail  No    —  fiita) 

=  -T   -45". 

I.et  ;-,  ,   ;.,  be  the  radii  of  the  outer  ami    inner  peripherics  uf 
the  wheel         Then 


''1  _  oa  ^  sin  ofa        sin  ))ifa 
r^       of       sin  oaf  ~    sin  oaf  ~ 

since  the  anyle  oaf  -r-  ,aiii  —  ma/  = 


^'"'      ,  -45°) 

'I"  —  .»•  \' 

sin  1     -^-  -  450J 

0  —  .r 


-45 


lii 


1,5'' 


fOKM    (V     .(,'  (  A/  /'. 


Ii< 


45     4- 


''.  +  r, 


'in 


tail 


tail 


an  I'ljuafion  <;i\in"'   t 


ri 


K-  i...mt  ,.■  in  uliKi,  ilu     |..r|u-n.li.ulai  ,>  /  f, 


I"'r[>rn,lH-iilar   ■  ,Mn  ,/;//  is  tl 


1  '"•  iiuit-  tin 


!<■  I  tlltlc 


IK'  in\  iil.ir  ari 


•  uul  ,'  / 


iicil. 


•' '  I-  til. 


Mi:riii)i)  II     !■ 


•\^\ 


]>ro(liK(,-(!    take   f//-  =  ,)/.       \\ 


to  no  (iocrilu-  tlic  ai*.  <>! 


lis   ■     '//,/,/'  ;-      I 

I  a-  icntrc  aii'I 


in    th 


'I  urclc  intt-rseetiii"  tl 


<•  I'xmt   /       J, ,11,   /•/,  „/,  .,,,,1 


o  .    anil    in    via 
I    lailiii-  equal 
ii    iiiiu  I  piriplu  r\ 


and  (//-/are  e\  uicntly  cqn 


'"■        I  In-    f\M.    tnaiiL'K  s   ,,,,/" 


tl  in  (  \  (  t\ 


ri-pi  A  ;.  and  tin  iilnn    tin 


Fi.:    J4(l. 
aiiKl*'  /v//  I-,  <-(|iial  tn  th 


•r...  2,40. 


anj^ks  t.)  ,ik  and/'   taiuH-ntia!  t<>  tl 


f    angle    ,./,/        Dimin-    ,/,      at    ri},'ht 


"^, 


r/(=  kat  -  90") 


U-  peiijilieiy  .it  /,  the  aiij^li- 


IS 


fiinal   iM   the  an},'le  o/,?  (=  ,</", 


anil   tliercfnrt-  "  ,1  =  c»/.      Thus  ,-    1-,  the  >.  nii,   ,,|   t 
arc  roquireil,  and  oa  (=■  op]  is  tin   laditi,. 


,/  _  90-), 

le  tircuiar 


.S7r/<.7:.S. 


137 


Ml  UK  ill  111  ,  I'i-,  :'4,(,,  I.rt  tin  Led  \mi1i  ,i  si,,],,  r,(.  s;u  , 
1  in  lOcxtiild  til  the  iminl  ,,  .111(1  tlli-il  1)(  in.idi  i  i  uu  riitric 
\utli  the  wlui  1  till  .1  (li--t,iii><-  ,,  -nl.ti.-iuliiiL;  .iii  .iili^ii'  (if  7,0^ 
■It  tlu-  (Ciiln  lit  tlu-  wind.  !,<t  till  iinan  l.ui  r.  li.ilt"  \v;i\- 
lKt\U(ii  tlu;  -Iii|)m;_;  liril  .illi!  tin-  -.irt.uc  i  •!  tin  .idv  .itK-ill"' 
V\.it(r,  -^tn'kc  tlir  mitc,  pii  ipliti  \-  at  the  pi. nit  '  \h:\K  fk- 
ni.ikiii;,;  ,iii  .lii-ic  nt  J^  with  , /,  .md  take  //•  c(|iial  ii>  mif  /la/^ 
<ir  -1  Mil  lentil--  (it  t!u'  a\ailalilc  tall.  <  i-  tin.-  i^-ntrt.'  .it  t.ic 
Liriular  ai\    i\i|!iiic<i.  and  ';'  i-  it--  ladin-. 

7-  Sluices.  I  In  Vi.iti:  ]--  r.ircly  ,iii|iiittcd  tn  tla  ulu-il 
witllMiit  -iniic  -lnh(_'  ,iM,iii:,;cnu  I"  uIikIi  iii.iy  t.ikr  tlic  ti  .tin  cf 
■  111  ii\cit.ill  --luiic  (I'i};.  25O), 
.111  nndiitliiw  --liiuc  Vv^.  .^51), 
'  II  .1  hiu  kit  111  jiipi-  sluice 
I  I-  ig.   2  V)- 

I  lit  |n|H-  -Ink  (•  i-<cs|)L'ciall\' 
.ld,i|)t(.d  |i  r  ,i  \  .u  \  iIIl;  ^iipph', 
In  ihl;  pi  1  .\  idid,  |i  11  .1  (.1-1  t.ini 
vcrtu  11  di--t,iiKi-.  with  ,t  series 
III  "hi  'It  t  ulic^,  -.(I  iiu  liiK  d  as 
til  i:i--ia  I  th.it  tlu  \k  .it(  I  intors 
tlu-  w  lu  I  1  111  tlu  I  iL;ht  dnci- 
tmn.  1  .ikinj;  i''-^  .1  till  in.  an 
(.  iictlu  u  lit  I  ■!"  h\  di.r  lu  u  --i-t- 
aiui  till  tlu  SI  tnli.  -.  tin  held 
Ji,  r'.-(|uirt'il  til  ]iiii(hKc  the 
\  I  lot.  its    ■  it  riiti  ,iiu  I'  ;■    is 


and  It  /'/  iv   til,-   t.ital  .ivailahlf 

(all. 

J/  -  /r   ^  //  _        '       '■■' 


=  rcinaindir  «»f  fall  avail.itiK   f.  i  iircssurt-work. 


4,;s 


s/nct-';. 


I 


i 


''"     pi'.t'ilc'    .//.'    in    :w    M\,Tt,i!l    ,ui,l    ,m     'uul.y,],,.,     -,1„k  o 
-h.Mihl  i,,nni,lr  vMth  fl!,.    M,!!-.,!,,,!;,-    iuth  <)*■  thr   luuvst  ■.ti.-.uii- 

''"'■-^    "*"    ''"■    J''l  llf    ^K'-t    ..(     Il„-    MVcrl.llI    .Il,,l,!,|    hr     pp.IHTlv 

Mirvcl,  ,iiul    tlic    mnci    .•.l-r-.nr  the   ini.ii-rtl.u   ..p.-nm.;  -houl,! 
lu'  cii  chill)   i-.)unilr(l  M)  ,,.  tM  I'limiii.uc    looses  due  t  .  cm!Uimc- 

tl(l!l 

1  he  iin.K-rllM\\    .liii,  .■-..pciinii;   sIkmiI.I  ,iU..  lu-  iiMfiiMl  to  the 
■ivi^  of  thr  H'l 

l.<t    /'.,    In-    till-  hr.ul    ahnvT  tlu-  eic-^t    ..t'  .111    .nnl'.il!    .hiur 
Then 

f',  ixin-  th.  ui.ltii  ,>(•  tin-  eiv^t.  .iikI  ,  tin-  cne«>k-iclU  ot'  dis- 
charjic.  I  lu  u  i.ltii  A,  i^  usually  3  <>r  4  i„s.  Ics.  tli.m  the  wkltli 
r  of  tliL-  wiiccl. 

I'Vom  tlii->  ciiu.itioii 


1         ' 


•""'   ''"    '''I't''    "'  ".'t.,    ,.\,,    th.-  cic<  n,    ill,  ,\   uMially  alx.nt 
«>  in-. 

A-iain.  t    >■  luaii  //,  ,  -   (  />i  rLciuirccI  t.i  produce  the  \ilocity 
r ,  at  the  jxiint  c<f  cntraiici-  A'  is 

II  ,-  - 
CJ)  ^  //,  =  '  . 

I  o  :;  i,*^ 

10  |K.T  com  ijcin^;  ailoMcd  for  Ins,  dut   t,,  irutinn 

Thus   the    liei^'ht   of  the   crest   ./    above  /,',    the    |>,„nt    oj 
entrance, 

=  A/)  =  i  /)-  CA  =  //,  _  *, 


SLl  ICFS. 


4.^9 


lUil  /.'J 


i>  a    iMr.ih'ila  'A  ith  its   \cr 


tr\    at  .1,  and    tlicrcfd 


if  w  is  tlic  an;4li'    brtui'm    flu'   lidri/on 
y.  /  to  tlu-  ])aral)i>!a  at  /.', 


tal  /.7>  and  tin-   taivrcnt 


AN 


,1/) 


AV 


lO  2i 


-A' 


3(?     \l 


'I'lic  luMi!    i\ailal)K    lur   prcssurc-wor 


k 


^    /)/■    =    /■(,    rr    //  //, 


I..t    i>    \)r    tin-    an;^I<-    i)it\\vt'n   /■' /    and  tin     taili;cnt    In    tli 
liccl's   iH-ri;i|ici  \    at    /•         1  In  n 


W  z-    til.    aiivlc  /:<'/■' 


/.'('  hcm^;    tiu-    ladiu-   to   the  (.(.ntic    "t    tlic    \\1)cl'1    and    0/'(, 
vertical 

If  till'  l(i\M'^t  I"  lint  (i    I  if  till'  w  lu(  1  jii^t  (,  Icai  s  till-  tail- 
thr  lir.'d  avail, d)l'    U<i  \ 


race 


iri'Ssiirc-w  I  iri- 


=  //       //,   _-  /■<,'        (H.    -  (U- 


/.'( '/■• 


Sciri};  the  radin-  t'l  tlu-  nufcr  pcriphiiy  nf  thr  vvhccl. 
if,  a'',\in,  tin-  watci  •titir-  t!ic  u  lurl  taia'i'nlialK  , 


t)\at 


a  —  II,   and  tin    >iii.d( 


//         //. 


/.('/•  ---  h, 


If  the   sluici'-i)|H-nin^j  is  •<>!    it   llu    \i  itcx  nf  tin     parahob 
till    a\is  (if  till-  i)|>iiiiii^  shnald  bi-  tani;ciUial  tn  tin    |iai  ihwla. 


•Jio 


fiRr  rsr  h-hfiis. 


'11   ^   to  .So  CU. 
I  i   tt.  C)   in>;.,  ,111(1    rarclv 


8.  Breast  Wheels.     These  u.ecl.  ,uv  uM.allv  .ul.^,,te,l  r,„ 
'■'"^  "'  "'"'"  -^  t"   i>  ft.,  ,,„,!    r,,r  ,,  ,l,,livcr>-  .,1'  m 

l!.     ])1T    s,'i   ..111], 

I  In-   .l..iniet.  r    sh,„il,l    l.c   .It  h  ,i,t 

"''^^■'■''"-    -'    ■'  "'■■    ^'i-^'tv    w,     ,,rt),e    «luel^    p,.,,,,lH.rv   i. 

i^.ncr,.lly    t,,.„,     ,^    „.    „.    ,    ,,.    ,,,,.    ^^,^^,,„^,^    „^_,    _^^^^^^    ^^^^^^^^ 

avriML^r  v.'I.Hitv  l.,.|n:^  ,ilK,ut  .jj   tt.    per  see.uul 

■'■'"■  ""I"'  ■■'tl"-^lH-.l.l„M,i,ln.,texcec.lfn,n,.M,.  mtt 
It  i-it;;,v,it  nnpM.t^iiu-e  tn  retain  the  u,,tr,-  ,„  tlu-  u  h.        i 

I"":.,    .'^     pn-Mhle.    .n.,1   th,.,     ,.   efCerte,!   either    1,V  IMtro.kumL;   the- 

"■'t«'>-  at  the  uniu    penpherv,   l,^.   ..53,  „r  by  sum.un.h    '^  the 


S»'' 


'*  • 


pi;-,- 


Ml 


fi-    - 


"1^ 


fi'i-  253- 


F). 


=  ?■)■ 


watcr-aiT    uith   an    ..pmn.    or   a   ,  nrh.    ,.r   .,    /■;,  ,,,7.    J.-j,,     ,,_^ 

which  nM>    he   e,.nstru>ie,l  ,,t   ,„„he,,  ir-n.  ...    stone      "l„   this 

case    the    iMuket.    ,n,u    l,e    pl,„K    il.  Mts,  as  the  c  uri.  rot.iins  the 

7'"-';'"'  "••■^  ^''-1''  •'••  -'■'<-'  •nu;h  ...thepcnphm-..!- 
tl.r  uheel.  „.  .,s  i„  rise  nut  mI  the  water  uit!,  the  least  rcMst- 
atKe 

Uluels  uitli  curbs  arc  (K-sJKnatcd  as  /n,:/i  />n.r^/,  hrast 
or  /.r,.-/„:ast  accrdinK  as  the  water  reache.  the  uhccl   near 
the  summit.  mi.l.IK-,  ,.r  button.,  uhih-  if  there  is  ,,0  curb  they 
«re  t.  rme.l  overshot,  mi.hllo-sh..t.  ami  un.ler.hot.  respectively 

""    'l'|"Ii"fa  float  sh.nihl  not   be   k-s  than    ?  x   ft      .,„| 


SPFFn   Or    ll-Hlhl..  .,j, 

tiK-  i..uv  iKiw.v,,  tu,,  .,,nsc,.  ut.vc-  float.  sl„,„l,l  In.  (UK.,!  to  -.t 
'■■•'■^t  ""^-i'^'il.  ."Hi  .VM,  tot,,,  tlnnU.  .,f-  :,.  ,a,,ac,tv  Tlu- 
lu-a.l  ..ncasarv.l  Iron,  .till  water  ovrr  the  sill  or  ],,,  .h,,„i,l  i,.. 
ahoiit  ()  ins. 

'Hlr  play  Int.      ,-„  the  outer  r>l,L;e  of  the  Ho.;-  ,„„!   theMiri. 
van.     from  ,]  in..  ,„  the  I.e^t  construe  te,l  nheeU.  to  j  ,n. 

The  <li.taiU-es  hetu.  en   the   llo,,,.   ,.  hon,    I  '.   I,,    ,  i  Xun.-.  the 
liead  o\er  tlie  sill  foi    sl,,u    ivheeK,   ai 
whtcls. 


'-■   '""i  ;i  'ttle    nioie    for  (lui.  k 


l^u-ast  wheels    are   an.on^-    ,,,,    !„„i    .,,    |„,,,,„,„    ,„„j,,,.^ 
havm-  an    edu  ieney  uhuh    ,n  ,v  l.e   as   .c;reat   a.   S.,   ,„,•  ,,,„" 
Ihe   .■mc-Rau:,    1^    usually  about   ;o    ,,..■  eent  lor   a  tall  of,,|„,„ 
>!  ft.,  ai)(l  50  per  ei-nt  lor  a  fall  of  4  ft. 

9.    Speed    of    Wheel.       The    .ater    leaws     the    l.uekets    and 
flous   auay  n.    the    rae.-    mr',    .>  vel,,e,ty    not    s.nsibh    .htlerc^t 


•  I      2  "5. 
fV..n,  theveloeity/.oftl,ewluel,   uhuh.   in   ,„a>tue,   ,s  usually 
about  .;/.■  Z-,,^- of  tl,evH,Hit>;|,'J  uitl,  uhuh    .he  ua.er  enters 

llh     wheel. 

I-ct  /i  bt  the  width  o(  (1,,.  ,\|,L.c|. 

Lot  .r  he  .leplh  ol  ,1„    wat,  ,    in  thr  lou  est  bucket. 


M 


,r- 


MUCH.-IMC.U.    l-IIFCr   <)/-■  HKI-.tsr    HHI.I.IS. 


A  lieu  in::  !nr  the  thickness  of  the  bucket'-,  tlic  play  between 
the  \sheel  ami  curb,  etc., 


F'  J. 


m 


m 


I-  beint;  ;in  eniiMrical  cdtTtuietU  whose  avera^'o  vahio  is 
.ibout  .9.       1  lelice 

10  0 
u  =  ' 

9  i>x 

<^         Q 

\\.  pr.ictice  /'  IS  i.tteii  taken  to  he         tn  It  is  important 

'  I :;      -  3 

that  /'  --houhl  hi  as  --niall  a^  ])ossibU-  and  hence  1  ^lioiild  be 
as  larLje  a^  |io--Mbl(  ,  its  \-,ihie  beiriL;  usu,ill\'  I  A  ft,  to  :;  ft. 

h  nuist  be  li.iine  in  mind,  however,  tli.it  any  iiuaaMsc  in 
tile  \ahie  ol  ■  Mill  i  ,ui--e  an  mi  rease  in  the  \\eii.  ..  of  water 
lilted  li\'  till  bviket--  .is  tlie\  eiperi;e  h'Mii  the  r.ice,  .md  \\<\\ 
therefore  tend  to  diniini--li  tin    elticieiuy. 

10.  Mechanical  Effect. —  Iheoretp  .dly  tlutota!  mechanical 
effect 

//  !)einL,'   the  fill  flora  tin     sm't.uf  of  -till  water  m  the   >iip[>l\- 
ch.iiinel  to  the  '.iirfne  ot  tin     v. iter  in  thi-  tail-i.ice. 
Tin--,  liowi'cr.  i-~  reduced  b\   tli(    ioHouiiv.;  losses: 
(a)   ()win""   to   frittional    iesj-t.iine.  cti    ,  theie    is   ,t  los^   of 


he. id  in    the    siippp,  -ch.imii  I  \'  hu  li  mav    he  me.isured  by  i- 

f  lieiMff  appro\im,itidy  .,',,  to   ^\^. 

Tlic  hcail  requiretl  to  produce  the  vehnily  .■,  ,it  entr.uicc 


=  (I  4- »'/''*• 


-*«.•   , 


if? 


MHCH.-IM(..ll    /If-li'li)/    HKl.tSI    IfHIHIS. 


443 


{I' 


LlI  ,iI  .    I 


];_;.    J;'>.   npii  -iiil  III  (iircLtKin  .mil  maL^nitiuio 


til 


c  \cliicit\  lit"  tin;  w  .Iter 


IMltlTIIlL;   till     hiu  Isl  t 

!  .I't  ,/./.  in  tlir  ilniL'ti'  '11 
"t"  tilt-  tan-c-iit  !•.  tiic 
wIk-c'i-  ])cii|)hcr_\.  icprr- 
^iiit  the  ".L'locity  II  'it  tlio 
]HTii)hi  \\  m  (lini.  Ill  III  .111(1 
ma^iutuilc-. 

COmjilrtc  the  p.ir.liklii- 
j,'rarii  /',i.  I  luii  ,;/■  r\  uUiiily  n  pix-sriit--  the  \iliiut\-  /'(it'thc 
w.itir  ic'i.iti\cl\  til  tile  wiiccl  Tills  xi'liifitv-  /■  is  r,i|iicl!y 
dc^triiX'fil,  the  Liirn-spiiiiiliiiL;  luss  nt'hcail  lieini; 


Pi...  2--U. 


'    '       ir  -t    r'l''  —  2nr^  cos  I- 


U) 


r  Ih  itii;  thi-  anisic  (/di 

.\<^\\m\\v^  th.U  the  water  i  ntci  .  tlic  r:u  c  \Mth  ,i  velocity 
f(iual  to  II,  till-  sprrd  nf  the  wlucl,  the  t hfi nit K al  useful  work 
p(  r  po'.iiui  of  uater  ])cr  seniiul  iliu    to  iinp.ict 

■I  I   "* 

_  "V       '  *        //        II 
uhicli  is  a  inaxi!    uni  .iiul 


(;■,  cos  )-   -  ii\. 


|-  Cos*-    y  ir 

4.V         "    .<' 


when  ,- 1  los  r  =:  ya. 

Ill  practice  ;■  is  usu.iHy  .ilmut  ,?u  ,  and 

tile  ni.ixmniin  useful  work        ^'' 

•  iiiK  ■-pondiiiL;  I"  t!u    relation  .\ii    _  I  ;,•■,  .    or   //  -  .43V  • 

To    diminisli    as    imuh    as   ]),.sm|)|,     the    loss   in    shock    at 

ciUrance  ilii    to  the  dissipation  o(  the  entTL'V  —  in  ed<l\  motion, 

2 1'  ' 

the  direction  ,»/>  of  the  relative  \elocit\    /should  he  paiallel  Ut 


4-14  MrCHA  ^!LU    i  i  I  hC  T   ( >/•    HKh.-IST    HHI.h/.S. 


l! 


'itfe 


ts^ 


til.'  ami  (ir  laii-niti.il  t..  th.-  lip  n\  t'lc  l)in  !,<t  and  should  th.  ;i  - 
I'll-,    hr  a])i.r().Miiiati  1\  at  ri-ht  aiiijlfs  to  tlic  \\1ut1^  pcni'lu  ry. 

II.  at  tllc  ])nint  (.Irntraim',  th-  iiikl  hp  is  thr  lovu'-t  point 
<>t  the  1)111.: ket,  the  \\  atrr  tlow  ^  upxi  ,ii(is.  ami  thr  rchiti\e  \ehicit\ 
/ ',  iiistia.l  ot  /■,/'<-  uholl\-  (l(stro_\iil  i:i  e.kl_\  motion,  i^  /<,/,- 
//■r?//V  (Icstroyjil  li\-  ;.;ra\it_\-.  'I  hi-  hittir  i-,  a-ain  restored  to 
the  water  on  its  rtti;rii,  an<l  iiKnas(-,  the  wiieei's  elliueiuy. 

lor  a  L;i\eii  -peed  (;<)  of  t!ie  wheil.  the  \ih)eit>-  (:,)  with 
whicli    thi-  water  should    reai.  h  ihe  wheel  in   order  to  make  tlit 

l"s-^  of    ^^    a  mmimiini  is  found  1>\-  making;  <//'       <>  in  e,j.   (ij, 

and  then 

O  —  r,  .  //r^  —  ,v  cos  r  .  ,/*, . 

^^^  ."•,    =   //  cos    )-. 

riiis  is  an  inipossihle  relation,  as  it  makes  r',  <  /.-■  iind  the 
u  :M  work  iie-atue.  In  la.et  the  an-le  .vA/ (  -  /:,,/■  in  s,k1i 
case  would  he  ,,o  ,  and  the  dueiticqi  ,//  of  ,  ,  wonM  I,,  prac- 
tiiaily  tan-t  ntial.  s,,  thai   no  watc  r  wouul  entei"  i^''  wIulI. 

A.L^am,  lor  a  -uen  \eh     il_\   ,  ^  olllie  water  as  it   leachies  the 

uheel.    the     speed    of  \\  heel    wlmll    WoiiM     make     the    loss   of 

J  "" 
a  mminmm  is  -i\ ,  n  l/\ 

p  —  //  .  ,//t  -  r-,  eos   y  ,  ,ii,, 
or  I,  —  r'l  cos  ,/ 

Iliis   is  also  an    impossij)!,'    r.latioii,  as  it    m.ikes  the  useful 
work  ////. 

''  "ill  '"■  found  a'Uantai^n'ous  to 
u-<-  1  ui\ .  d  oi-  poly-on.il  hiKkets  and 
not    p|,,ne    lloals.        \     hiieket,    for    c\- 

•"'M'l'  ■    'iii>     toiisi^t   o|   three     tr.iii^jit 
^       ixirtion.      „/,     /,,  ,     ,,/,     |,j^,      _,.-.        ,  I,- 

tliesc  the  inner  portion,  ,,/,  shouKI  he- 
radial:  the  outi  r  poition,  ,//',  |,  ne,irl\- 
normnl  to  the  p,  ,iph.  r\-  of  th.  uh,,  1.  .m.l  <he  centr.d  portion, 
bf.   m.w  in. ike  .in,L;li-s  of  .ih  ,ut    i  ;;     with  ,//'  ,\\\k\  .,/. 


Flu.  257. 


P:: 


sjw.iiifi. 


Mlit.i  I. IMC  A I    n-ll-CI    Ol     HRI  .-isl 


li'Hri-IS. 


I)i- 


rcLianiiivj-   a!l    dth.-r    I, 


tllC    U  lu'cl    ill    toi,t-I      )1H1(N 


'-■■^L'-,.    t!i;'    tiicorctica!    uc\ 


445 


ivtTV    of 


^vI^■ll■     h.   — 
\(.'lcH-it\    v.. 


=  wQ 


t.it.il    t".il 


\  u  V,  COS  r  —  ui 


g 


4-n. 


r,ii 


H'.jiiircd     t.i    111". 


lULl'       tl 


le 


If    ;/     he     tin-     ct"! 

'I'm'-  cxpciiinnit- 


li."lCIK\-,     tlR'Il,     ,K 


tile    rc-^iilt-^    of 


V  -  .40  to  .45  it  //_  ^_     //. 


tf=^  .4-'  t..  .4j  if/  -//; 


V  =  .47 


'T/^  -    "//; 


V  = 


33 


if//,  -   V/. 


T 


u-rr  i-  ,1  io,,  ol  li,/,i(!  i\uc  to  tiK  ti.iii.'i! 


tin-  ch.mnci  m  wIirIi  tl<i-  wheel  work,. 


11, il  le-utaiKc  al' 


\ -et  /  =   ! 


en-til  .i|  tl  e  ehannel 


eir  I  iirl)>. 


i.et  /  —  tliuLiie--,  ot  uat.-r-l.n  er  !( 


I.ct  /' 


lireailtti  ol"  w  heel 


■I  1'-;    tile    uhi-ol. 


1  lie  me  Ml  \eloeit\    01  il.iu    m  tl 


ii-    euil)   ilianiiel 


imateU 


1--  approx- 


n.  ,u\d  till   loss  i>t  litiiii 


<!!>'    /,•  ,  liaiuhl  fr 


iiti,'ii 


^    t 


I'  -4-  2/  VW  ,        4  ./■  4-  2t 


ht 


-X 


.?  />/       2r 


ul 


icrc  /  =  cocfficif 


Ml  \    ..I'  liKti.in,  /' 


/'/  —  water  area, 


->-  -'/  uettei 


'/'  'n- 


aiiil    )■  !)em^  30' 


piritneter. 


err  IS  a  In-,,  ,,t'  h 


e,u'  due  to  til 


the  end,  and  side,  o|  t 


le   l)lieket> 


e^eape  of  watiT  n\(r 


I.et 


tli<-  pla>    between  tiie  end.  o|  Hu'  huckelsaiul  lli 


liamii  I . 


44' 


v;/ ' // 7V/(   ;/    //.'/:(./   ()/   !ih'i:.isr  ifnii-j.s. 


!H-'  r 


Iff  J! 

1*^1    i:l 


;j-. 


f 


;i 


I  flit   ^ 


1^ 

p  ^iw 

I-ct   .s  ,  In.'    tlu-   |ila\    .It   tile   -^iiir-..        (.\|    -:   ,\,  ,    ,ij)])l'(i\illl,iti,-!\'. 
I-<--t     .|,      ;._,,    .    ,   .    „     he    tlu-    'Irjitlis    nl'  u.itcr    ill    a     hucki't 
e  ()rrt--|)i  mdlllL;      ti>      /.'     -lU  ir--^i\  c      pi  )-.|tlnlu      i:i     it^ 

(Ic-^icnt  frniii  till-  ni,ri\inL,'  t"  the  lnwc^t  point 
I, ft   /,,/,.    .       .   /,    l)c    the    I  •  M  ii  ^))i  iii<lm;4    w  atn-arc^    iiicas- 

iiird  aIniiL;  the  \'.  lif( T-.  i>t,Tiphi'i\'. 
I'ltr  MiilRr  111' cii-.cliar'^;c  at  riii!  nt  a  l)iiiki,'t    —  />s 
1  lie  iman  amount  ol  u  atcr  (--tapini^  frniii  a  hiickct  omt  iti 


f/'.v,  V 


--  ^'~ 


*-. 


I    Ix'illL;'  the  C(n  fluiriit  of  disi.  liart;^. 

I  lu-  ualcT  (.'--i.  a]ic--  at  the  sides  as  o\-cr  a  scries  of  weir^. 
and  the'  mean  amount  o)  \\  ,itei-  e-i  apniL^  from  a  hneket  over  the 
sides 


X  ,' ^,  ♦  -:v 


/,  v.r,  4-Ai-,-f  .  .  ,  /,  i^. 


Heme  the  total   loss  of  efk  ^  t  from  -si  ape  of  watc 


♦-.s7)^^i*'^,  f  ♦■^=•1- 


..) 


])er    see-,   /'    heini;    the  \erlii  al    ■li^taiue    hi'tvu-en    the    point    of 
entiaiua.'  and  the  --iirlaee  of  the  wati'r  in  the  tail-raee 


C'l    'I'liere  i-.  a  lo^-,  of  head  dm    to  ionrnal  friction. 

Let    //  uclL;lit   of  w  lii-el 

l.ct  re,     :  %vcML;ht  ot  \\,|t,  r  on  the  wheel. 

Let  I  ^  r.idiiis  o|  wheels  outer  ])erniher\-. 

1 .1  t   /  r.idin^  •  A  .i\lc. 


i-r.  a  iii  K 


\ua:h.ish:.u  ihia.i  oi  itRi.isr  uiihf.is.  447 

Loss  per  sicoiul  of  incchaiiic.il  ftlcct  (int.-  t"  journal  tViction 

-  /'(//■+  tr,  I     I,. 

r  hciiiL;  tlu-  KidTicicnt  t^'i  jmirn.d  Irictioii. 

I  luTc  IS  ,1  |(]^^  nC  iiu-i  li.iniil  cllijct  due  to  tlio  rr^istancc  nf 
the  .iir  to  tlu'  iiioti.m  ot  tli-  il.iats  Jjuckcts  .  hut  tlii-  i>  prac- 
tically very  small,  and  ina>-  he  disrci^'ardod  uithnut  scnsihlc 
riror  A  dci'prnm-  of  the  tail-race  jiroduces  a  furtlier  hi^-  of 
edect,  and  should  (iiil\-  lie  adoj.ted  when  hack-water  is  feared. 
IliiUi    ill,   /i'!tt!  (II /nil/  n/ii/i(iiii(  ,1/  ctfiit.   jiuttiTi" 


is  =  w(J[  //  - 


:<'0 


'"'\  ^■"■^  y 


(7'  ■         //"  -!      •'  ■ 


^''  t- "',,   // 


■■/     ,    -'VC' 


f  J  ""  1  I''  '  ' 


r.'C-Y 


/-I   2/4 


:.■// 


/' 


--"■Of    \       \:j~  u    '/•+:.■,/.'''• 

MeiK(      fni     a     .L,'P,:n     \alue     of    ,- ,     the     niechauKal     effi-ct 
(oniitt  n-  the  last  ternii  is  a  nia\iiiuini  ulien 

.  ,  Cos  y 
"  =  2     '^-   -4.13   X   .-,  .   if  1-         ^O'l. 

In    praitic.    th.-    sp,-,,|  ,  ,f  the  u  heel    i-    made  aho'it  ,v/,    /,,t/r 
«.f  th<'   VelocitV   Ultll    V.  hi,  h    the    uater  enters   the   w  heel. 

lor   a    .;ivMi  sp,,-d    n     uheel,  and    disre-ardui-   the    loss  of 


MICROCOPr    RESOLUTION    TEST    CHART 

ANSI  and   ISO   TEST   CHAPT   >,        ^ 


1.0 


I.I 


•^  I--    12.2 
[2.0 

1.8 


1.25 


^      APPLIED  IfVMGF     Inc 


4vS 


i:\.iMrt.i:. 


flCi'Lt  line-  tn  ^>irl)  frirtiup,   wliicli  i.   aluays  small,  t!u'  mcJi.m- 
ical  cKcct  is  a  niaximum  tof  a  value  tA  :■    L,M\t-n  1)\- 


*    'i 


■Is      "; 


II  COS  ;' 


(■  +  4-^{?N^ 


1  lie     lo---;    tiy   (.si  a])(.'    uf    \\  atci  , 


Z 


,    I    t  J^'     ,   \an(s,  (,|,    ,111 
// 

avi'ia-r,   Ir.Mii    into   15    [ht   cent    ..Ithr  uli.,lf    mi]  pK  ,   -.,,    thai 

<■  ♦  a.i,'-     vanes  from    '    to 

"  10        20 

l-x.   Tlio  liiKk.'ts  (,f  a   l.nv  breast   wlic.l.  o(  J4  It.  .lianiPter,  arc  half 
fille<l  with  wat.r  wh,.  h  (lows  from  a  flume  throiij-h  a  vertical  rectangular 
sluice-openiiiK  at  the  late  of  1;  ni.  ft.  per  seror.,1      Tiie  Hnear  speed  ,,f 
the  wheers  periphery  is   5  ft.   per  seeoeul.     .\t   ih-   |.ni„t   of  admiss,,,, 
Iheuillowmi;  ict  has  a  velc,.:ily  of  lu  It.  per  seconil  ,,nd  iii„kesaii  .,tiL;l.-.,| 


'^ 


tzA 


Fl.:.  2<;S. 


^o    w,th  the  run.     The  total  avadahle  fall  ,s  8,  ft.    F.nd  (,n  the  position 
o      he  potnt  of  admission  ;  ,.,  ,he  work  ..o„*   ,.y  i,„p,;,  ,,"  l^; "^  " 

..  •'     ►^""  •■""''"""•""«"««(  the  slu.ce-open,„K  the  .lep.h  of  ,|  ^ 

snroiidinj;  being  ij  ins.  "^pwi  01  uu 


V.I 


S.-tOhHlhS    H  HiiEt.  4  4'» 

(<j)  Let  Oil  he  the  radius  to  llu-  point  n\  admission  li.  .mil  Irt  'P  \it 
its  111!  lination  totlie  vertical. 

iJraw  the  vortii  al  ' 't/  ami  tlu'  li'irizontal  HI'. 

Theoretical ly,  /h  .  llie  liead  required  to  develop  a  velocity  of  lo  It.  per 
second, 

'"' 
-64='-"- 
Thi-ii  SJ  —  I  ,\  =  6iJ,  ft.  =  head  availahle  for  work  hv  weight 
—  the  vertical  fall  on  the  wheel 
=  FlJ. 

(}j.-       .     515 

Tlirrrfore       cos  </>  =  ^^^_,  =     "      -  -  =    4-^18-5. 

and  ip  =  65    3',  defiiiinj;  the  position  of  />'. 

(//I  The  useful  theoretical  work  done  l>y  impact 

6''' 

=    "*  15  .  51 10  cos  30°       5)  —  f;-/).  n3  ft.-lbs. 

Tlie  useful  theoretical  work  done  by  wi  11,'lit 
=  62J  .15.  6{;  =  6503.906  ft. -lbs., 
ami  the  combined  useful  work  —  7040.039  ft. -lbs. 

((•)  Li  t  .//>.  Z)'/.*  be  the  vertical  and  li.)ri/ontal  distances  of  the  lift 
/•/  from  /)'. 

The  aiigie  between  the  diie<tioii  of  -']  ai  A  .■  id  the  horizontal 

=  0  -  30°  =  35*  3' 


Thcri'fore 


and 


.l/>  =■-  ^^^  sin' 35' 3'  ==    51533  ft- 

/f/^  =  ,      sill  70'  !•'  --    1.461;:  ft. 
^4 


1.5 


6  ft  . 


.\n.iiil,  the  width  of  the  wheel 

and  the  widlli  of  ilie  sluice  may  be  taken  to  be  about  3  ins.  less 
than  tliis,  or  5I  ft.  The  heail  over  the  lip  -  1,%  —.51533  =  ■047^; 
the  average   vrlnrity   of  flow   thioiii;!!   the  sluice  i  V''4  ^  1.047^  = 

7.3656  ft.  per  second,  and  the  depth  of  ihe  sluiccopenint;  ^ 

^  .154  ft. 

II.  Sagebien  Wheels,  I  i^;.  259,  havo  pl.itic  floats  iiulincil 
t<>  the  radius  at  from  40'  to  45^  in  the  tiircction  of  the  wheel's 
rotation.  The  floats  are  near  together  and  sjnk  slow!)-  into 
the  fluid  mass.  The  level  of  the  water  in  the  l1oat-pass,ij,'es 
jjradiiaily  varies,  and  the  »'>ltniie  dischar^;cd  in  a  jjiven  time 
may  he  very  ^jreatly  chanj^ed.       Ihe  t-fficiency  rif  these  wheels 


u 


:     1; 


i    :  li.. 


450 


oyr.RSHin  h-hhel 


is  over  80  por  cent,  and  lias   rcaclud   (.ven  90  per  cent.      Tlie 
action  is  almost  the  same  as  it"  the  water  uere  transferred  from 


_/ 


I         ■ 


the  iii)iKr  tu  ilu:  lower  race,  without  aijitation,  frictional  resis- 
tance, etc..  flow in.L,' away  witlioiit  obstruction  into  the  tail-race. 

12.  Overshot  Wheels.  -Since  the  introduction  and  develop- 
nunt  wi  the  turbine  thcH-  wheels  have  l)ecome  .almost  ol>solctc. 

riu\  h.ive  been  considered  .inion^'  the  liest  of  hydraulic  motors 
for  lalis  of  S  to  ;o  ft.  and  for  a  deliver}-  of  3  to  j;  cii.  ft.  per 
second,  .md  h.i\e  proved  es|)ecial!y  useful  lor  l.dls  .,f  ij  to 
20  ft.  I  he  efficienc)-  of  overshot  wheels  of  the  best  Construc- 
tion is  from  .70  to  ..S3. 

1  he  thicktiess  of  the  sheet  of  w.iter  passint;  throu_L;h  the 
sluice  on  to  the  wlieel  rarely  exceeds  4  or  ;  in-.,  and  is  often 
less  than  2  ins. 

If  the  level  of  the  luvul-water  is  liable  to  ,1  ^^reater  v.iriation 
th.m  :;  ft.,  ills  most  ad\antaH;eous  to  emi)loy  a  pitch-back  or 
\\\'^\\  breast  wheel,  which  receives  the  water  on  the  s  inie  side 
as  the  channel  of  ,i])pro,nh 

13.  Wheel  Velocity.  -  I  his  evidently  depemls  upon  the 
work  to  be  done,  ant!  upon  the  vclocitv  with  whiili  the  water 
arrives  on  the  wheel.      Overshot  wheels  shoul.l  h.ive  a  low  cir- 


oii:KSH(yr  h'hi.el. 


45» 


ciimferontial  spectl,  \ar_\inL;  from  ki  ft.  por  sccoik!  Inr  lar,L,'e 
whci'ls  to  3  tt.  per  sccntu!  for  small  wheels,  ami  slioulil  imt  t)e 
liss  than  2\  ft.  jjcr  scloikI.  .At  a  liiL;her  speed  th.ui  6  Tt.  ])er 
second,  i''  the  buckets  are  mere  than  /:,•.-'  t/iir,/s  full,  the 
efFKienc)'  does  not  exceed  Oo  per  cent. 

In  order  that  the  water  ina>-  enter  tlie  inickets  easih-,  its 
\elocit_\'  should  he  L;reater  than  the  ])eripheral  velocitv  of  the 
■heel. 

14.   Effect  of  Centrifugal  Force Consider  a  ninlecule  of 

weight  .\-  HI  till    ■•  unknowi!    '  --urface  of  tlie  water  in  a   bucket 
(Fig.  260).      .\t  eaeli  moment  tliere 
is   a  d\nami(  al  ecpiilibnum   between 
the  "forces  "  acting  on  ;//,  \  1/.  :  (i^ 
its    weiglit    ri'i       2  I    tile    centrifugal 

force       (.*■';•;   13  1    the    result. int    /'  of 

.s 

tile  neighborin;;  reactions. 

''"ake    .)//••  z^  :.:     MG  ^  "''..//•, 

and  complete  the  parallelogram  J-d. 
Then  Mil—  /'.  The  direction  of 
/  i-,  of  course,  normal  to  the  surface 
of  the  water  in  the  bucket. 

Let  //.I/ produceii  meet  the  ver- 
tical through  the  axis  <'of  the  wltt 
111  I..      Then 

74' 
M<1  _  /""''  _  FH  _  O.U  _     /-_ 

.!//•■  ~    :.•    ~  J//'"  "  ''/      ryr 

and  therefore 


0   .T   ,) 


^f3 


Vic.  260. 


01  = 


2<>i5 


<u' 


ft., 


tiikmg  .i,'  =  3J  ft.  and  ;/  being  the  number  of  revolutions  per 
minute. 


• 


f  i 


Mf 


452 


Oyi:h'sH(U     H-UHlil.. 


iif:  m 


I   i 


li 


.« 

V 


'I  lu;s  tlic  po-itioii  n|  /;  is  i!iilii)(.-in!cnt  (it';-  .ui<i  dfrlic  pn.-i- 
tinii  (^t  tlic  hiukit.  ^.1  that  ,ill  tlic  ii..rin,il>  h>  the  u  atn--sur!;;cc 
111  a  hiakct  iiu-i.t  in  /,,  .md  the  surtaci.'  is  tlif  arc  nf  a  tirclc 
haMiiL;  its  centre  at  /: .  or.  rather,  ,i  c>lin<h-ical  surface  with 
axis  thrnu,L;h  /;  ])arallel  to  the  axis  ( .t' n  ■tation. 

15.  Weight  of  Water  on  Wheel  and  Arc  of  Discharge. 

Let  (J—  \i.llinie  --iipplied  per  see,  .ili(l.\'--   Ilillllher  of  buc  kets. 
1  lien     -       ^_   luinihir  of  jjuckels  let!  per  second. 


2T 

2  -T  (J 


-     \oliinie  ol  \\ater  recei\eii  hy  each  Inicket  iier  sec. 


Hence    the    .u'e.i    occupied  1)_\    the  u.iter  until    s])iUiiV4  coni- 

mcMces  —         -  .  i>  hcinj;  tlie  bucket's  wiilth  (        width  oi'whicl 
o.\  (»; 

between  tlu'  shrouilin;4s  i. 

The  water   ilows  on    to  the  w  hi-c  1  t!ir.iu;di  ,1   ch.umel     I'il:. 

261),  u-nall>'  ot' the  same  wi(hh  /■  as  the  wheel,  .uid  the  -uppl>- 

is    reL;nl,ited    by  ine.uis   ot'  .m    adui-^tabh'    shiiee,   winch  in,i\    be 

eitlu'r  wrtic.d,   iik  lined,  or  liori/untal. 

When  the  w.iter  spim^.s  clear  I'roin  the  shiiei^,  .is  in  j'i^-. 
2'i|,the  axis  ot  tlicshiikc  sh,,nl.ll„-  t.iiii;enti.d  to  the  ,ixis  of 
the  let,  ,inil  the  inner  cil<;cs  ot  the  shiice-ojieiniiL;  sh,,uld  be 
rounded  so  ,is  to  elimin.ite  comr.utioii 

Let  r,  -  be  the  lioii/ont.d  and  veitiial  dist.uues  between 
the  shiicc  and  the  p.iint  o|  entr.mcc. 

Let    /  bi   the  time  ot  tlow  I)ot\vccii  the  slnii.e  .uid  entr.mce. 

Let  .„,  ;  I  be  tin  velocities  of  t1ow  on  le.iviiiL;  the  sluice 
and  on  (.■iiterinrj  tlie  bucket. 


and 


■'i  cos  (^'-f  (S) /'  =  _)■, 


-»=7V4- V". 


'1  —  ' ,. 

rf  bein-  the  ,mj,'iilar  d<\i.ition  oCtho  i)oiiit  ofenlr.uui'  from  the 


wm 


orr.RsHor  lynEH.i. 


4x? 


suniniit,  and    ;'   tlir   allele   between    tlu  ilirccti m  <if  motion  of 
the  water  and  tlie  \\lieel  ;it  the  juiint  of  entrance. 

If  the  bed  of  tile  ehanned  is  lidri/Mntal.  and  if  aNo  the  sluice 
is  vertical,  openinij  iijiwards  from  the  l)ed,  and  is  of  the  same 
width  l>  as  the  wheel,  then 

Q  ::^   />t  \  J.^//,  , 


Fie.,  afii. 


/  beiiiL;    the    de]itli    of  shiice-nju-nini;    and  I'l ^  the   effective  head 
ii\er     the     s 
actual  head. 


ii\er     the     sluice.         Ihi^    etiective    head    is    abt)ut     i^,,    of    the 


riiii'-.  t.dNin;,;  .;         },2,  -r  —  8///,t  j;ivcs  the  deliver)'  per  foot 

width  1 4'  \\  heel. 

lakin^'    .*'.   ft     and    ^-,.6   ft.    as   tile   extreme   limits   between 
which  //.  >li<'nlil  lie,   and  .j  ft.  .uid  .;;  ft.  .i-- the  e.\treme  limits 


4i 


»f 


4:-« 


:wr;r.>'-'r  h'hi:h!.. 


bLtuccii  whiLh  /  slinuld  lie.  then   "  niii-^t  lie  ])ct\\ccn  tlic  limits 

(' 

I.J4  aiul  5,  aiul  an  aNX-ra-c  value   (.f  J  is  3.      'I'lius   the  uithh 

ot  the  uiieel  shduld  lie  11,1  the  avera'a    ^    "". 

.\;^ain,  iliMe-anhn-  tiie  tliickness  of  the  buckets,  tlie 
capat  it\  r,|  the  portiMii  ,it"  the  wlieel  passini;  in  fmnt  of  the 
water-supply  per  -eKiml 

_  -    s>\'    «'•,  -  "'•I-'  /      ,  ,  ,       '/.      ,  , 

—  '"''-.   ^   —  ,  ,- = ''^""''^^'•,—  ^  y  =  wr,'-'.  approximatel\-, 

30 

r,  liein--  the  railius  and  u  the  \rloeit\-  .if  the  outer  t  ircumferenec 
ot  the  wliecl,  ./  the  .lepth  of  the  shroudniL;-,  or  erown,  and  ;/ 
the  uuniher  of  1  e\  ohition-,  p,r  nnnute. 

<»ii!\  .1  i».rtion.  houe\er,  of  the  space  can  he  occupied  !)>• 
tile  uatei.  s, ,  tint  the  c  apacitv  -if  a  bucket  is  ,itni>,l.  ui  bemj; 
a  trattion  less  than  ,init_\  and  usuall\-  ^  or  i.  i-'or  ver\  hi'^h 
u  lleels    ;//  nia\-  be  | ,       I  |i  ik  e 


Ik  re  fore 


Dibdu     ;  0. 
tmiu  =    r. 


?£ 


\^t 


1  he    delivery   \A    per   foot   of  wi.lth    must    not    exceed    .1 

lertain  hnu't,  otliemise  (  itlier  ,/ or  .7  will  be  too  j^reat.  In  the 
loriiier  case  tlie  water  would  acijuire  too  i;reat  a  veloLit\  on 
entering,'  the  buckets,  wliicli  would  lead  to  an  excessive  loss 
in  eddy  motion  and  a  correspondin;.;  joss  of  efficiency ;  uliile 
if  the  speetl  //  of  the  wheel  is  too  j^reat,  the  efTicienc)-  is  a^'aiii 
(liminislied  and  niij;ht  fall  even  l)eiow  40  per  cent. 

'llic  di'iith  of  a  bucket  or  of  the   shnjiidinj;  varies  from  10 
to    lO  ins.,  beiny  usually  from    10  to  12  Ins.,  and  the  buckets 


^m 


oyhRSHor  iJ/Hhi-.i. 


4?;> 


arc  ^])rea(l  ainiv_;  tlic  nutfi"  circuiiik  rcit' c  at  iiUcrx'als  nt  I J  to 
14  in-.  Tlic  lumibrr  <■(  the  hin.  kct-  i-  a])])!'' iXimatcly  5^,  <>i" 
Gi^ ,  /'i  bciny  in  ttut. 

The  ttTicit  ncy  iif  tlio  w  lu-cl  nfccs-aiilv  incrca-v>  w  itii  tlio 
nunilnr  of  t!u  bucket-,  hut  the  nunihi.r  i-  Hinited  1)\'  en  tain 
ccinsiilerations,  viz.  :  ui\  the  bucket  tliickness  mu>t  not  take- 
up  too   niueli  ot   tile  wlieel-   iieripher)' ;      in  the  nunilnr  of  the 


t  > 
I     \ 


FlO.   2f)J. 

buckets  must  not  be  so  ^ri'at  as  to  ob'^truct  the  free  entrance- 
of  the  water:  (<''i  the  form  of  the  b  cket  es',entiall\-  affects  the 
iiwnilier 

i.et  tlie  bucket,  I'i-^.   2'')2,  coiisi>;t  of  two  portion-,  an  inner 


45<> 


Ol'FKSHO/     H'HIitL. 


I  his    '  i  i 


l%-  ■  "  'i 


M 


!'^ 


M 


]M.rtioi)  /',-,  wliivh  i-,  ra.li.il.  ,mil  an  mitcr  portion.  <,/;  c  hcinj^r  a 
ponil    on  uhat    i.  calk-.l  the    .lixiMoii    circle.       The   Icni,rtli  /v  is 
usiMlly  one  h.ih  or  two  thir.l.  ,.f  the  .lepth  ,/ of  the  .hroudnvr 
■]  ake  /\-         I  /. 

It  ni.iyalM,  he  ,is,uineil  \vithout  nuicli  error  th.it  the  w.iter- 
surface  „,/  i-^  ,ipproxnnatel\-  perj)en(hcuLir  to  the  line  , ,/.  m, 
that  the  ,m-ie  ,,/,!  i-  appro\ini.itel>-  a  ri.L^ht  an-h'. 

Ihe  --pilhn,-  e\i<leiitl\  connnence.  ulien  the  c\-h'n,h-ical 
surface,  liavin-  it--  axis   at  ,   an,i  oittm-   .,n   Iro,,,    the' bucket  a 

2rr() 
w.iter-are.i  ciu.il  to   ^^^^.-^,   parses    throu-h  the  outer   ed-e  ,/ of 

the  bucket, 

I-et  li  he  the  Iniiket  a-i.-K'  <(  V. 

Let  "  \)r  the  iiK  hnatioil  of  ( ',/  to  tlie  lioii/,  ,11. 

I.ef  (,'.  l)e  the  inchnatioii  of  „,/i,,  the  hori/on 

l.et  /■,  lie  the  r.uliu^  of  the  outer  periphir>-. 

r.et  A'  he  the  radiu-^  olthe  .h\ision  circle. 

Let  /.  be  the  raduis  .,tthe  inner  peripher\- 

Ihen 

g    _   Oe       cos  I  "  ±  ,,">) 
r,f.r  ~  Od  sin  o ") 

th.-  M-n  Inin^;  phis  or  minus   acconhr.-  as  the  bucket  is  below 
-•'■  •il'"\e  the  hon/oni,,i.  ,uh!  in  the    latter  case,  li  H  ^   ,;,,  tlien 
'\'^'-  --  .i,''  sin  «,'). 
A;j;aiii. 

11/  =_A/tan  1"  -J    ,;.  ,  api)roxiniatel\-. 
1  hei  (-lore 

tl'^  •"■'-■'  '(^"         \   Urn  :^  i-  ,;,\  :-^  '['  tan  («-[-  0), 
where  ,/  —   ;-|  _  /■_.       ]\^■,^^:^■ 

the  .irea  ,,/',,/  :.  area  ,W  -  area  /',./-  area  ,0,i 


r,R 


-in   li 


'"•'     ,  ^  2  70 


■Mt 


wri 


Ol-'LK.SHO/    HHF.liL. 


457 


Equations  (ij  and  fJi  .L^ivc  n  and  0,  and  therefore  the  pnsi- 
ti'in  (if  the  bucket  when  spiHinir  commences. 

I  1h'  liucket  will  be  complete!}-  emptied  when  it  has  reat  heil 
a  p..-.,ti,,n  in  which  , ,/  i-  ivrpcndicular  to  a  line  from  <■  t., 
middle  pcim  of  ,,/,  ,,r,  appro\miatel\-,  whe:,  ,,/V  i<  a  ri-ht 
anqle. 

Let  w, ,  f/),  be  the  correspcmdin'^-  \akies  of  n  anil  r,'>,  and  let 
r,  be  tlic  an<(le  between  <,/  and  the  tan,-<nt  at  ,/  to  the  wheel's 
pcripheiN'.       Then 


and 


r,  =  9o°  -  (n^  +  0,}. 


sin  0,        r ,(«>-" 


two  eiiiiati.iiis  t;i\inL[  </',  ,md  ". 

Also,  it  </•  is  drawn  perpendicular  t<>  <>,/, 


tan  r,  r    cot  cdk  -     ,  ' 

'  '  ck 


R~nwr    -  R- cosec /?  _  cot /y. 


I  he  vertical  di-taiu  e  between  the  pi.iilts  w  lu-re  -piUinL,' 
be.uins  and  iiuU,  \i/.,  i  ^y>m  i\  sin  ";  can  now  be  d.ter- 
mined. 

'l"he  pitch-an-le  (  ^  ;/•  1  is  the  an-le    betu.'en  tw<,   consecu- 

360 


tive    buckets   so    that    »/• 


.\' 


In    order    to    obtain    a   small 


,dc    (=  J',)   between    the  hp   .,f  the  bucket  and    the    wheel's 


/>    .i,Meatcr 


periphery,    it    is    usual    to    make    the    bucket     aiiLde 
than  (/■. 

boi   exainph', 

(j      5    _  5  360  _  450 ' 
'    '    4         4    f     ~     N   • 

The  interval  between  the  buckets  should  be  at  least  suffi- 
cient to  i)rexent  any  bucket  di|.pinL;  into  the  one  beh.w  at  the 
moment  tiie  latter  bej,n'ns  to  spill. 


45^ 


OytRSHOI     IrHll.l.. 


'.ctiV'i-',   Fif;.  .''63,  be   tile   ciniMMii  .iii^^lc,  and  ^  the   thick- 
ness of  tlic  bucket. 


Th 


en 


Fii;.   263. 


ff   =       —       tan  i^i  -\-  ,p\  =       tan  ^, 


approvimatcly,  and  theref<ire 


„/  d 

N^r,/^  +  t  -     tan  >i]  =  irrr,.       .     .     .     (3) 


'^k 


N 


''13» 


AIsii,  by  e(|uation  '  2  i, 
r,R 


r,  d-  2-tO 

in  li  --  ^/i  tan  «+    J^. 

2  2  '   bNoj 


.      (4> 


Tl  jse  la--t  two  equations  L;i\e  .\'  and  ^. 

1  he  nunilnr  of  buckets   max   .dso  be   aj>i)n)\iniately  tound 
from  the  formula 


a 


15.   Form   and    Capacity   of    Bucket.  —  In    practice    tli 

huc'Ki  t    may   be  liehlle.iled  as  toiiows: 

In  l•■i^^  2G3  let  M'  —  distance  between  two  buckets. 


CAP  ACHY   Oh   liUCKET. 


459 


5,6,  ,/ 

Take  (fd"  ~-     dd     d    ,/,/';   ;ilso  take  in     -     .  and   ioin  r/< 

4  3  2  •*  ■ 

'I'hi-;  i,Mvc-i  the  ti>rm  of  ,i  suitable  wooden  bucket. 

If  tile   buekut    i>   <it    inm,  (.ircular  arc  i'^  substituted    ("V  tlu. 

])'  illi(  )T1S    /'(  .    ((/. 

ALjain.   let   /w,    Vi'^.    2C14.   he    the    th'aknes-,    of  the    stuani 
just  befi'ie  enteriuL;'  tlic  bucket. 


Kk;.  264. 


Let  1///  l)e  tlie  tliickne^s  of  the  stream  just  after  enterinjj 
the  bucket. 

Let  )-,  be  the  ani^le  between  the  bucket's  lip  and  the  w  lieel'.s 
peripher}-. 

Then 


mbdii^  —  cap.'.:.t\-  of  bucket  =  b-t\  .  pvi  =  bV .  dn 

=  b:\dp  sin  y  =  bV .  dp  .  sin  y, , 


and  therefore 


dp 


wd»^  tiidu^ 

r'l  sin  y  "   I'  sin  y^' 


Now  ovcrshtjt  wheels  cannot  be  \entilated.  and   it   is  con- 
.sc(iuently  necessary  to   leave  ample   space  abo\c  the   entering 


460 


c.-ip.-K.ii  Y  or  HI  (:ki  r. 


strrain  |..r  the  free  exit   (■»'  ;ui .      Tliu-,,  iic-lcam-    ll.iat    thick- 
ness, 


tlic  lh•^t.ln^^■  hrtwicii  c  1  m^ccutivr  flo: 


Its 


=  rA/    (l-'i^.  2('>-\)    ■  ,//> 


I  '  sill    I-,' 


and  .\\  the  luiniber  n!  lnukcts, 


<  - 


2nrJ'  sin  ;' 


»/<///. 


I  or  ffficient  aiti'.n  the  tuimluT  ut"  tin  hiukits  is  miith  los 
tlian  the  Innit  i^iven  In-  this  nlalion.  ..ften  not  exceedinj;  one 
lialf  of  siicl)  limit 

II  ;',  is  very  Miiall.  /  '  =  7,  —  u^ .  appioximately,  and  there- 
fore 

2n/  ,  sill  I- 


,^...,.,i,-,-^_     . 


<:m 


t^ 


!  hr  I  apa.  it\  of  a  tuh  ket  dci>eiids  ujion  its  fdrni;  and  tlie 
hiieket  iiiu-t  l)e  so  ilesi-,'ne(l  that  tlir  uatir  c  an  enter  iVeeiv  and 
witlioiit  sliock.  is  retained  to  the  lowest  possible  point,  ami  is 
fmallv  di-eliarfjed  without  let  or  hindrance.  Hence  ilat 
i)ucket-,  I'ij,'.  265,  .ire  n,,t  so  cflfkient  as  the  curved  iron 
!)ncket  in  I-'ijr.  26.S  ;ind  as  the  loinp.pimd  bucket  made  of  three 
or  two  pieces  in  I-"i^;s.  266,  2(>;.  ,uu\  2(»).  Ilie  resistance  to 
entrance  is  least  in  the  curved  bucket,  as  there  arc  no  abrupt 
chan-es  of  direction  due  to  an^,'lcs.  The  capacity  of  .i  com- 
pound bucket  m.i\-  t)e  increased,  without  diminisliin-  the  ease 
of  entr.uice,  by  ni.ikin^,-  the  inner  |)ortion  strike  the  inner 
periphery  at  an  acute  an^de.  I'ii;  .'r„>.  1  he  objection  to  this 
lon-^truction.  esiK-ciaII\  if  the  rel.itivo  velocity  /'  is  Inrfjc,  is 
that  tile  water  tends  to  return  in  the  opposite  direction  and 
escape  from  the  bucket. 


"^ 


C.-fF.K.Ijy    Ol-    LiLLKhJ. 


41U 


Fii;.  2(13. 


Fk;.  266. 


Fig.  26- 


VsX.-- 


Fig.  a68. 


Fic:.  269. 


Fta.  tfO' 


462 


c.-^r-iciTY  Oh  tin.kh.T. 


I.c-t  /'I'll,  c/x .  I'ii^.  -70,  riprcMjnt  t\\(j  consecutive  buckets 
«t  an  .ivci-slint  wheel  turnin-  in  tlie  direction  sliov\n  by  the 
arrow. 

Water  will  cease  to  enter  tile  bucket-space  between  /-<,/ 
and  ,■/[•;,  and  ini|)act  will  therefore  cease,  when  the  u])i)er 
parabolic  bouiular>-  of  the  snpijly-streani  intersects  the  ed-e  ,t. 
'llie  last  Huid  element^  will  then  strike  the  water  alri'ady  in 
the  bucket  at  a  point  .1/,  whose  vertical  distance  below  ,/  nia\ 
be  desi.Linated  b\-  ;.  Ihe  velocity  ,-■,'  with  which  the  enterin.i; 
particles  re.ich  .1/  is  -iseii  by  the  eipiatioii 


'   ! 


'  '  I      I    -A  ^• 


I  I 


Aj,'ain.   while  the  lluid    particle-^   move   from  ,/ to  .]/ kt   the 
bucket-  nioM-  into  the  positions  tVi'b  \  ff'i;'. 
I.ct  arc  (///    —  ij  =:  cc' . 
Let  arc  liM       s.. 

Let    /  lie  the  lime  of  mo\  enient  from  ,/ to  ,/    ;,,r  ,/ to    I/) 
Then 

s,  ■--  u  T 


^i 


fc4 


s.,=  -^'   '^Z, 
2 


assuming'  that  the  mean  velocity  from  ,/ to  ,1/  is  an   aritiimetic 
mean  between  tile  initial  and  tinal  vejocit)' i>f  entrance.      Thus 


■,  +  •• , 


f2) 


Also,  since  the  an},Me  between,/.)/ and  the  wheels  jK-ripliory 
is  small,  it  nia\-  be  assumed  that 

the-  arc  liM  ^  ,/,•  -f  ,/-f  ,■,••,  apprDximately, 


'I"llU<5 


E\  -IMPI  rs. 


4";, 


WfF 


2ni\ 


'-f  ^-i 


(3) 


and  by  I'qiiations  '2,  and  (^3). 
'  \  1  „ 


2 II 


"'1    ■, 


an  equation  .Ljivin^-  aj)i)r..ximatLly  tlu  di-,taiKx  v,  passed 
tlir-ai^di  by  a  lloat  durin-  impact.  iln;  buckets  can  now  l)c 
plotted  ill  the  positions  tliey  occiiiu'  at  the  en<l  of  the  inii)act. 
The  amount  of  uat.r  in  <acli  bucket  l)ein.-  also  knoun.  the 
water-surface  can  be  delineated,  an. I  henci  the  vertical  distance 
z  can  be  at  once  found. 

I..\.  I.  IituI  iliij  angular  (le|.rcssion  u(  liie  water-.surfare  below  the 
li..nz<.nt;il  <..)  wher:  ih.-  hm  kt-l  lip  is  37'  14' above  tlio  centre,  and  yh) 
when  tlie  biKkft  lip  IS  on  a  levi-l  with  the  centre;  also  lind  ,0  tin- 
posiiiin  of  the  bucket  below  the  centre  when  a  horizont.il  thrdUKJi  the 
hp  bisects  the  an^le  bttwecn  the  wairr-iirface  anti  the  radius  to  the  lip 
The  whe.'l  has  a  dianut-r  .<f  3;  ft.  and  makes  loj  revulutioMb  per  minute! 


The  aiii^ul.ir  vcl.icnv  r.) 
(.1) 


44  I"* 


I  I 

id' 


3J\ioy 


sin  «,'»  121 

cos  (37*' 1 4'  —  05  ~  J3o" 


Therefore      .2°°  ^  «^"»  f 37*  « 4' -  ./>l 
131  >in  O 


rot  <p   -   \  ■},\^> 


(*) 


\Zi  _         >in  «,'» 
200~  Ci>»  (»  -I-   0) 


t.iii  ^'> 


I  r 


■bO), 


and 


w 


iil  SI  I    o 

200  ~  cos  (0  +  01 


<>    -  31"  lo'. 

_   «iii  0  ^        i«i„  0 
C.i»  ;0  ~  I   _  J  ^j„i  0' 


or 

and 
or 


•Ill'  0  +      -  sin  0  - 
111 

tin  0  =  .405,S. 

0  =  23*  5b'  =  h. 


f 


464 


liXAMI'l.ES. 


li^, 


Kx.  z.   All  ovcr'ihot  wheel   has  a  (li.iiiieier  "f    32  ft.,  a  1 2-in.  croiiii 
and  Its  peripheral  ^jjccd  is  4  11.  per  sec  .mi.     The   hp  .-f  tiic  uuckn  is  , ', 

MIS.    thick.       Water     enters     tin- 
■  wheel   ill    a  direction   inclined   at 

1  60'  t.)  the  vertical  at  a  point   iz° 

30'  Iron)  tile  Mimniit  and  with  a 
vclicity  of  10  ft.  per  secnd. 
Spilling' <-(.iniiiences  ai  :2o'tri'm 
tlic  summit.  Iiml  (,ji  the  relative 
velocity  ( I  )  at  admi.ssioii  ;  (/■)  the 
anj,'le  l)etween  the  horizontal  ami 
the  water-surface  at  1°  47'  3-,' 
al).,veand  at  30°  below  the  C(ii- 
tre;  if  1  ttic  an^le  (>':)  between  tlic 
bucket  lip  and  the  rim:  (,/ 1  the 
point  where  the  bucket  is  emp- 
tied :  (cl  the  bucket  ant;le;  (/>  the 
elbow  an^le;  i_i,m  the  number  of 
t)iickets;  (// 1  the  bucket  water 
area. 

.\t  the  point  of  admission  li 
let  .(^'//  be  the  triangle  of  veloci- 
ties so  that  ,/i;  =  16  ft..  ,/A  -  4  ft., 
and  the  aii^le  i;i//i  ^  30'  —  12°  jo' 
=  I ;°  30'- 

Assiimini;  also  tliat  the  water 
enters  wtthoiil  shoik-.  the  telative 
veloritv  r  =  (,^)  is  paiallel  to 
the  bucket  arm  c<i,  and  the  aii^le 

Ciil-^  y,  =  anglci'//^-, 
Ihcn 


t^.^ 


1\ 


(a)  /••  =r  >»f«  =  ,^»  +  ,//.  .    2.  .  c-/ .  <M  cos  17"  30' 

=  16'  -h  4*   -  J  .  16  .  4  cos  17*  30' 
-  149.9342. 
and   /■  =  12.2443  f'.  I'LT  sec. 

[b)  When  0  =  t°  47'  33", 

33      _  COM  I"  47'  33"  -  <!•)  _ 
l6(,\?~  sin  ..-.^        ^  ~  -^^• 

or  cot  0  =  32  sec  I'  47'  33"  -  tan  i"  47'  33", 

and  ^  -  ,•  ^7'  33". 

When  0=30-.     «^"»'3o\+   0) 

■'  sin  0  J^' 


t 


or 

and 


U') 


or 


HXAMni.S.  4'>? 

cot  </>  -  •52  ser  30    +  t:in  30'  =  37.5-77. 

!,'>  =  I"  31'  24". 

sin  17'  30'  _  1 2.244  "> 
sill  y,  10 


12244^ 

rosec  r,  =         ^        i.iscc  17    30    -  2.545. 


i!i<i  ;'i  =  23'  ^'• 


Ul)  At  tlie  |Mpiiu  ./i,  \viicr<'  tin;  spilling'  is  (onipiftrd.  (><<!-<■.  is  a  riglil 
allele  and  tiie  aMt,'lc  i>ih<-  —  angle  t\,UjX-i  =  ;',      Ttien 


sin  </>!  _eilt  _    r,  _  '^/_£\V_    ' 
sin  r>  ~  ^  "  .C   ~  32\'6/  ~  3-' 


w 


sin  <>i  — 


and 


Si 
./.,  ^  o   42'. 


.01227; 


Tlierclore  ''1  —  '^'  —  ir^   ^  '.'■i  1  ~  ''16°  lo', 

and  \\m:  bucket  is  cmptii-d  at  '/J     ♦^  ''•     lu'  =   1  j'.*  10'  from  the  sunimit. 
(<■)  r,  —  il/;     A' =  15V.      Tliertforc 

[(,  —  \z,\  ,■■  s  f! 

i.iii  23°  s  --  t.iii  /i  ,         ,  .4-72. 

■^  1  3I  sill    /J 

Tliis  last  <.-fiii.iUmi  IS  iMsily  rediiii-d  ti)  tin-  furin 

IMS'  fJ  —  I.745S  Cos  n        —  .74'-'74. 
and  »"s  li  =  .99^12. 

or  /i  ==  5'  «'. 


121 

1350 


in  ( ircular  measure. 


(/)  The  elbow  angle  Oci  -  t«o°  —  fS  -  ( W        18   -  /J  -  .,^'  _  >  , 
=  90"  -  5"  «■    f  23    « 
=  loK°. 


(A'' 


../  121         I        I  \       44 


or  X  —  79.8,  say  80, 

An  empirical  approximate  rule  makes 
ztir,       44      l*^' 


.\   = 


,( 


ioo|. 


466 


h:.\.tMrLi:s. 


\\ 

ill    %  •    : 
11     I'  \ 


O 


sS 


N 


{/') 


>.\\  ;^  ^    — 


I  21 


I 

^  tiiii  Jo  +  water  area  of  Ijiicket. 

I'litTcforo 

the  water  area  -.  11.09475   -  "-'O.-ijjJ  -  .2SH67 

=  .72275  .-q.  fl. 

~  104  sq.  iiis. 

;-:x.    V   ..\n  over>hot  waler-wi.eei,  of  40  ft.  ,|,an,eter,  ,s  .2  i„.  wide  a,Kl 

...s  a  ,.,-„,,  ,„n.u,l,„,.     Tf,e  p,tch-an«le  ,s  4' and  ti.e  tinckne  J  of  Ihe 

^.  k,t  ,,,,  „  ,     ,        ^,  ,,,^  ,„„„^  ^^.,_^^.^.  ^^^.||._^^^  conunenccs  the  bucket 

«au.,  area  ,,s  ni  s,,.  ,„s.      F„ul  the  n.unber  of  buckets,  the  point  where 

^pMu.^  .•onnnenees.  and  the  an.Ie  between  ,he  rnn  and  the  t.uket  lip 

I  0.6 


♦■i    -  20  ft.;      /•  _  JO 


I'aki'  li.  thi'  bu(  ki-i-ani;;e   ^-  '  4' 

4 


19'J  ft.;     ;a  _  JO  -  .S  .^  !9.2  ft. 


Til 


.T     •         1^ 
1X0  1; 


>1U 


tan  0 


19.2 


Ib-nrr 


""•^'V    5 
2  i  ,So 

A--  .64.5, 

'  tan  0  --  J.  56.      ..r     ';    -  f,S'  41, 

1  be  empirical  forniuja  j^ives 

2»'i        44  20 


t.iii  0 


-4  J 
'44' 


.V  = 


,/ 


i;7. 1. 


Ak:, 


10 


J 

f^  eoscc:  5    -.   cot  5'  _    .2778;. 


ami  V, 


I  5°  V''. 


I.x.  4^  '>„,./.'/.»//,  oi  the  theoM.tic  capacity  of  .,  ,„„,,,  „  „||ed  with 
X  . Iter.  I  he  an.le  between  the  bucket  hp  and  the  wheels  penpherv  is 
•o  the  rad.us  ,0  ine  oim-r  penpherv  is  ,«  ft.and  the  depth  of  the  crown 
.s  .2  „s.  If  he  velo.;.ty  of  ,he  «,,ter  at  entrance  ,s  Lr.-  that  of  the 
wheels    penpherv,  lind   the  ^^reaiest    numb.-r  o,    buckets  theoretically 


rile  nuinlKT 


-'T»-,  sin  yJ7>,         \ 
2',' 18  sin  20/  \ 

~7— (^-'j 


<  roj.  2. 

The  1.  t„;il  n.imber  i.iav  be  about  l>»..  tliinis  of  i|,,s,  or  69. 


2    ■    .!    . 


mi-.(:h.-ini<:-ii.  F.hhucr  of  oi^FRj;H(->r  iVHi:t:i.s.  467 

16.  Useful  Effect.  -^  ,m  hfUct  of  U'a\i^'-/it.~-^'\\\^  uhcd 
•^IiMiiKl  liaiiL;  trrcly.  nr  just  Jc.ir  the  tail-u  ,it(.T  -urf.icc,  and  tin; 
tut.i!  I.i'l  i>  inc,iMi-((!  Iriiii  tlic  surtacf  nt"  the  water  in  the  tail- 
raee  lm  the  w  ,.t.T--,Lirta^e  jii^t  in  fiMiit  .if  the  sluices  thrmii^^h 
Mhieh  the  water  h  !>r'pUL;lit  nu  t.i  the  wheel. 

Let    //|  .     I'l-      J7J,     he    the    Vertical    ill>t,ini.e    hi  t\Teen     the 


Fi'i.  272. 

centres  of  L;ravit\-  of  tlie  water-areas  .)f  the  first  and  hi~l  Ijiickcts 
hefnre  --piniiii,'  cnnimence-..        I'iien 

h,       R  cos  ■)       r,  sin  '-.   very   nearly. 

Let  //,  l)c  tile  vortical  distance  between  the  (  <  ntres  ,if 
.i,navit>  of  the  water-area  of  the  bucket  wliicli  fii -^t  he^'ins  to 
si)ill  and  tlic  point  at  which  tlie   spi'.linj^   is  completed.       Then 

h,  =  r,(sin  ^\  —  sin  "1.  very  nearly. 

The  usehil  work  per  sec.  -  wQ  h  •  kh  /(  jtein^  a  trac- 
tion <  I  and  ai)proximatcly        .3. 

Let  A„  be  the  water-, irea  in  the  bucket  ^^hah  .irst  be^'ins 
to  spill. 


i 


P« 


\  " 


'  ,i 


■^ 


t^4 


•ii 


46S  Ml  iji.iMc.ii  iiifj'i  di-  i)r//-:sH<)'r  nHrris. 

Jk-t\ucii  tin--  biiclsct  ,iiul  tin  ^iic  whiili  i--  tlr^t  •■iii])tii.-(l, 
i.e..  in  tlic  \i.rtii,il  (li-.t,iiur  // . ,  iii-nt  .*  luie  kct'-,  .it  1  ([iial  dis- 
tances .ip.iil.   and  let  tluir  w  .itci'-.iri  a^  ./,  ,  .1  ,.  .1 /    l)c 

(■ar('fii!Iy  >. .  IcnlatLil. 

l.(.t  0„.  Ijc  the  mean  aniiMnit  'ii'uater  ]»r  hueket  in  the 
vhseliarLjinL;'  ,irc. 

I  At  ./,„  be  tlu  mi-an  w  .iter-aiea  per  bucket  in  the  <hsi.h,ir>'- 
iii'.;  are. 

i  hen 


-',., 


.7,,-^. /,-+-./, -4-  .  .  .  ^..J,    ,-  ./ 


1  hi-  \  ah f /•  1  .111  nw\v   be  i-.i<ily  li>miil,  -inei- 


k  .-- 


o       ./„■ 


Let  (/  lie  tile  varyiiiL;  amount  of  water  in  a  bucket  from 
wliich  spillin-  i--  takiiiL;  place,  .uni  .it  anv  incmt  nt  let  ;  be 
the  vertical  di-t.ince  lietween  the  miter  eil^e  'if  the  bucket  ami 
the  su -f.ice  (if  the  w  .iter  in  tlu-  t.iil-r.ic;-. 

1/  is  .1  fimctinii  lit  !  .md  depend-  ii[iiin  tlie  ouitnur  nf  the 
uater  in  the  bncki-l. 

I  .1  t  !  be  the  ///,  r///  \alue  nf  1  between  tile  jininls  where 
spilliiiL;  Ik  Ljins    uid  t.-niU.  i  i-.,  I'nr  \-,iliies  y   and  r.  of  y.       Ilu-ii 


'•(^S     '   / 


/  .'"''/  =-''i"v.';,      /  ■/  '('■■ 


since 


/.'     ''''/  ^.'V  -    / 


'/   •   '(!•• 


.\L:jain,  tlio  clemfp*..iry  tpi.tntit}-  of  w.iter,  7y,  havin^^  .,11 
initi.il  velocity  ecju.il  to  that  of  tlic  wheel,  vi/.,  //.  f.ilK  .1  dis- 
t.itue  J-  and  .iciiuiri  ■-  .1  velncitv         S  ir  -\-  J^w. 


*nnr 


MlAJi  -ISU'.^:-    lilih.CI    lit    (>r/:A'\//( '/     li'HI:HlS.  \<") 

'lliiis    it    lliiu-,    AWAY    ill    ilu-    t,iil-r;icc,  Laiwini;    .i    I<'->-    of 

vpillini;  l)c<_;iii--  atu!  cm'- 

()\rr-li"t  and  uiuli-haik  ulu'cl-  <\n  ii'.l  u  "rk  well  in  back- 
uatir.  a-  tin.-,  iitt  a  -rcatcr  .t  !o--  \vci:^^Iit  .)t"  w  atcr  in  n-iiv^ 
.it)(>\(    tlu'  ^urtacc. 

It"  tin-  u.'trr-lc\c'  ill  the  raci  i-  liable  t..  \ari.i';i(iii  it  i- 
iH'tlcr  t.i  diniini-h  tlu  dianutci"  ..f  tli.  uIuh-1  and  d.-M-n  it  -m 
th.it  it  nia\    iu\<r  b<    iiinm-r-<-d  to  a  L;r(-ati.r  diptli  than   I  J  nis. 

i/;)  /-V/'''"/  ''/  liiipiit.  'i'hr  head  //  ri'(|uiix-d  t<  i  prciduco  the 
vclni-itv  :•,  with  wliiLli  the  w.itrv  rt  aclu'-  the  wlicci  i>  thcoroti- 

.allv  "'    ;    l>i't   •'-   thriT    i-   a    hi—   i>f  at  Ica-t    ;  per  cent  in  the 

— .^ 

ini.-t  iierleet  de'iiNeiw  it  i-  ii-u,d  t..  t,ik<    /'/  v  ^  ^.  an  averaL;e 

'  '  -A 

value  (if  V  bein:,,--    1.1. 

Let   the  w.iter   enter   th<    Itucket    in   the    direetinn    ,;<  .   Vvg. 

-"t,        Take   </i  =  :  ,        The    water    w<\\    nmve^    rnuiid    with    a 

\eiout\    11     a--nnie<l   the    -aine   a-  that  .if  the   ,li\i-.i.  m    eircle  , 

.xw\  l.-.ue^  thr  wheel  witli  the  -anie    \.-li..  it}".        lake  ah  in  the 

ilnaeti.ni    ..t    the    tan-ent   to  the   di\i-'nn  eirele  at    the    p<iint  of 

entrance   r     u  The    ennip^nent    /■-     reprc-ent-    the    relative 

vel.H-ity   r  ot"  the  water   with    ri-jv  Lt   t. .   the   bucket,  and    this 

v<-l(Hity  i-  uhnllv   de-tici>-ed.      ah  niu-t   neccssarils   be  par.illil 

t..   the    "ulei    arm  -if  the   Inuket,   -..  that  there   ma\-  be  im  loss 

■lit  -hoJs  at  entrance.       I'lieii  th<    inipuUive  etiect 


i«! 


47°  MlAJI.IMCAl.    Hhhf-XT   (IF   Ot^'/rAVS/ycr/'    UHFHIS. 

But 

y   bfiii.L,'   tlu'  aii-lc   thr  'w^h  uliii.li    tile  \i  iter   is  dtviatud   from 
its  nri,L;in,il  (lirrctiMii  at  the  puint  i  if  ciitraiKi  . 


'  t 


U  V,  COS  !         u"), 


IS 


H, 


\^A 


ami  tlu'   1 1 1  i  \i    I  -^1  I  I  I    KKi-'KCT 


wQ 


wQih,;kh,i  uvcos:      u      loss  due  to  journal  friction. 


Desii^niatini;   tlic   fir-t   tu,,   trriiis  ,,f  tlii^   -  xpr.-.si,,n    l,y  /• 
Xhv  In^^s  (lur  t( .  iDiinial  trii'tmii 

=  «!■>"■!;". 

^j  hcini,'  tin-  radius  dt'th.-  .i\lr,  .m^\    //the  i\ei-ht  nf  the  \\lic<:l 


FX.-tMP!  E. 


471 


Ex.  An  oversliot  whrnl  weij,'liin^  2o,ocx)  lbs.,  with  :i  12-iii.  crown  amt 
of  40  It.  diameter,  receives  400  (  u.  it.  of  water  per  minute  ami  revolves 
ill  '-ill.  Ije  raid's.  The  water  nuers  the  Imckels  at  IJ°  fn>m  the  wheel's 
suinmit.  with  a  velcn  i:y  of  16  It.  per  second  aiifi  at  .111  ant;!.-  of  10"  with 
the  wheel's  peiipherv,  which  moves  wiih  a  linear  \eloi  ity  of  9  ft.  p.- 
seccjiid.  Spidin-^  coiiuiUMices  and  is  compiiteii  at  points  which  an 
respectivelv  140"  and  160°  from  the  wheels  snmiiiit.  Determine  the- 
power  o|  the  \.hecl  .uid  it--  clli.  ienc  y.  taking;  /!•  =  .  5  and  »  -^.04. 

'lake  J\   ~  radius  of  division  ciri  le  =   !■,;  ft.      I'lu-ii 

//,  =.   lyj  ((IS  I  J     +    ;o  cos  40    -    ^4,v>47<'()-  it., 

and  //,  =  20  cos  .:o"  —      :o  cos  40°  =  3.472964  ft. 

Therefore  the  !1.1'.  due  to  7i.'t-(^lit 


Uil  .  400/  I  \ 


=  27.37::'5, 
and  the  H.I',  uiu-  l,>  iiiip.ut 


bl\       400 

. 9(l6  cos  10° 

32     33CXXJ  ^ 


'J) 


=  l.43'K'8. 

Again.  liie  weii;li'  '  f  the  watrr  on  the  wheel 

400.  (12  J/  12S  1      2o\ 

= \20Tt   --       +    20.T  .       .       , 

bo  A)     \         l.So  .'     i8oy 

=  2231.04  Ihs..  .ippro.N., 

and  the  total  weigiit  011  the  a.\le  =  22231.04  lbs. 

Thus  the  energy  absorbed  by  frictional  resistaiK  e  111  H.P. 

2223;. 04  \ 

—  —     ,  —  X  .04  t  (j       =  .iSi8y, 
550  ^       •'40  ^• 

and  hence 

the  net  useful  Wdik  in  11.1'.  ^  27.37215    t-   1  43<)(iS  —  .1S189 

=  28.62994. 
The  total  av.iil.ible  H.l". 


62i  .  400  /  16' 


i3ooo    \64 


v^T  +  20  cos  I : 
28.62994 


20)  —   ^3.0022, 


and  therefore  ilie  elhciencv  —  ^^-'yy-  =  .867s. 

33.00223  " 


/!/(. //-/<;'. k   a/// 1:1. 


17.  .\  pitch-back  <■>■  high  breast  wheel  i,  t  .  he  i)rcieno<l 
to  ail  over-iiot  wheel  when  the  ■.iirf;iee-le\  els  ..ftlic  licad-  ami 
tail-watei-  ,ire  li.ilile  t"  very  k  ,ii-.iiler,il)le  Minatinii, 

III  th'  piteh-haLk  wheel  the  water  i-,  ailniiited  hv  an 
.idiustahle  ^lin\-i-  iiU(.  the  l)iieket->  (Hi  the  -aille  side  ,is  the 
supply-chainiel.  J"ii;s.  J74  and  J75.       Ihus  the  uliccl  revolves 


l-^ 


i.'tfe 


•^« 


ti::| 


i-'f:.    274. 


Fii;.   275 


in  the  dirceliitn    in  uliicli    the  water   leaver,  and   the    drounini,^ 
>f  the  wheel   i->  ])ie\-eiited.       I'urther,   the   hneket-    ina\-  he  iiou' 
\entilated,     I-'i-\     j-J.     and    nia\     therefdre     he     placed     closer 
t<')i,'etlier  th.in  in  the  nn\  entilati-d  M\ei-^h..l   wheel. 

I'hi'  emeit'iiey  ..(the    piteh-haJs  i-.  at   lea^t  equal  t. .  that  "f 
the  ov  er-.liot. 


tX.IM/'l.tS. 


473 


10(1 

iiul 


ail 
tllc 


/ 


EXAMPLES. 

1.  An  uniiersiiot  wheel  wc.rks  .n  a  rrctangulur  rhannel  4  ft.  wido.  111 
whidi  the  water  on  the  up-stn-ain  side  is  :  ft.  deep  and  (lows  with  a 
velocity  of  12  it.  per  second  ;  the  water  on  the  dovvn-strcani  sitlc  is  3  ft. 
deep.      Find  tlie  useful  work  done  and  the  etiiciency. 

.Ills.    1000  ft. -lbs.;   }'.. 

2.  1  >eteriinne  liie  nKi.xiinnin  mechanical  effect  of  an  undershot  wheel 
of  i:  ft.  dianietci  niakuij;  10  re\<>lutions  per  minute,  the  fall  bein^'  •,  ft. 
aiifl  till'  (luaiitity  nf  water  passed  per  secmd  1;  <u.  ft. 

./'/v.  1423  ft. -lbs. 
'.    .\scerl.iin   the    ooneial    pr.  •portions    of    a 
R'Acn  :  lieii;ht  of  fall  =  4.I  ft.:  delivery  of  water 
radius  if  e.Nterion  ircumference  =  9  It  ;  ;'       20' 
.his.    .<  -    143°    57':     </'  ==  \2S-.i;    d  - 


I'oiKelet    wlieel.  heiiis,' 

-  40  cu.  ft.  per  second  ; 

-  2    ft.  ;      '■    -    z.x-    v..  . 


\ 


1; 


5  ins.;  -V  =  57  ;  ?/  -  .(«>. 


4.  r)e«iun  a  l'wn<-elet  wherl  for  a  fall  of  4.;  ft.  and  24  ni.  ft.  of  w.iter 
per  si-cond.  usin^' the  fortnui.e  nn  piLjcs  42S  4',.',  takiiii,'  y  —  20',  tmd 
ul.so  A  =  ;o    as  a  first  appro,\ini.ition, 

.Ins.  .1  -  143°  57';  depth  of  iruwii  i.S  ft.,  d<'pth  <  f  stre.nn 
=  .372  f;  ;  /'  — 4.14  ft.;  radius  of  bucket  =  2.26  ft.:  (/■  ~  128"  f.'; 
\  =  17'  i';  number  of  buckets  -  4S  ;  mechanical  effect  --  S.5  H.I*.; 
etiiciency  -:  .67, 

5,  ,\n  undershcit  w;iter-wheel  with  str.ii^'it  Ihj. its  weighing  1  ;,ooo  lbs. 
woiks  .11  .i  sti.ii^h'  rcctani^ular  ch.inncl  ol  the  s.ime  wulth  as  the  wheel, 
viz..  4  ft.,  the  stream  delivers  28  cu.  ft.  of  water  per  second,  and  the 
efficiency  i^  \.  i'md  the  relation  bi-tween  the  up-stream  and  down- 
stream velocities,  if  the  velocity  of  the  intlowniK'  water  is  20  ft.  [ler 
secoiul.  find  the  velocity  on  the  down-stream  side  and  determine  the 
mechanical  effect  of  the  wheel,  its  di.imeter  beiiij,'  .••o  ft..  \\w  di.inuter  of 
the  ^adfjeons  bein^  4  itf..  and  the  ccjelficient  'if  fiution  .oo.S. 

Ans.  3C)34  o'')  ft. -lbs, 
(\.   Determine  the  elTect   of  a  low  !i,\isf  m  undershot  wheel  13  ft.  in 
di:imeter  .ind  niakiii',' 8  revohiti- ui-^  pii  minute,  the   fall   is  4  ft.  and   tnc 
delivery  20  cu.  ft.  per  second  ;  the  velocity  "f  the  stre.im  before  comini; 
on  the  wheel  ,s  double  that  of  the  wheel.  A'is.  3148  ft. -lbs. 

7.  20  cu.  ft.  of  w.iter  per  seciid  'iitcr  .ui  undcishMt  wheel  "f  30  ft. 
tlianietei.  makiiif;  8  revoluti'ins  per  ininnte.  tlirnu^h  ,u\  underflow 
sluice.  Till'  velocity  of  tin- enter  in:;  w.iu-i  1-.  twne  th.it  <>l  the  wheel's 
periphetv.  rmd  \,i)  the  head  of  w.itiT  behind  the  sluice;  U,)  the  fall,  (c) 
the  the'-retical  mechanical  effect  :  {d)  the  actual  mechanical  ciTect,  disre- 
gardmiL:  axle-fi  iction. 


■t: } 


f  x.-iMii  rs. 


mi 


i 


^  ./'/>.  Ul  2.7 '0  ft  ;   ,/,,  ,.,,S3  ft;  ,,  ,  5-,  „   ;,  .    ,,/,   ,,.^  ,,  p_ 

•  •  -o  <ii.  It.  nt  watrr  per  sf.oncl  onicr  im  uiulrrshot  w  hrol  of  ^o  ft 
Mu.nct..r  ,n  .  s,ra,«l„  ,ar...  ,hc  fall  bei„.  3  ft.     Tl.c  .l.,,tl,  of  ,),..  ..n,..,.' 

M.h  Ih,.  number  „f  the  floats,  of  wIm.I.  four  are  imuK-r^cl.  .s  48 
■'-'••-IMS  ■  ft.  long.  The  vv,.i,l,tof„H.ul,edis  7.00  lbs.,  the  nulms' 
"f  the  a..lc  ,s  ,  J  ,„s..  an,l  the  coCficent  of  frict.on  is  . ,.  Fi„,,  ,  /  ,. 
..est^speed  for  the  wheel  ;  ,.,  the  eorn  s,.o„,l„„  ...eehanical  effect  ;    ;!  Z 

v.l,:'!^'  V"  ''  I'-  '"•■'  T""'  •  '^>  '-3-  "•''••  assuminK  'lu-  speed  of 
wheel  re.lnced  ,,,  5  74  f,,  per  sccon.l  by  axlc-fr,rtlo„  ;  (< ,    ^\ 

V.  72CU.  ..of  water  are  del,ve,e,l  to  an  undershot  wheel  w.thsiramht 
loa.s    through  a  channel  of   re.  tan«ular  section  and   5  f-.  u„l...      ,"  ,! 

.  t"wi;:i'; ''- ";"""■? '"r '^ ^^ ''•"' "-^ ^''^■"""-  '^ theemcenc; 

e  r.        r      :     'i       :?■  ''";"  '''^'  '-■'!''>'-^-''  '<P-'  («)  -f.l.c  wheel  n.ust 
be  r,  U.  per  second.     .\|so  deternm.e  the  nu-chan.ral  eflcct  of  the  wheel. 

,,,    Ti  .       .  ■''"•  '01^5  ft.-lbs.  per  sicond. 

.0.  Fu  water  ■„  ..  rectanyular  channel,  of  constant  width.  ,s  .  Mt 
deep,  and  nnp.HKcs  upon  the  (i.,:  bucket,  of  an  undersho.  wheel  w.th  ' 
velon  y  of  „  f,.  ,..  sec.  Show  that  the  efHciency  ,.  greatest  and  e,  ,d 
to  .l.Si  for  a  pcKpheral  spe.'fl  of  S.S^i  ft.  p,r  se.- 

...  Water  enters  the  buckets  of  a  low  bre ast-wheel  w,th  a  velocv  •  f 
.0  ft.  per  sec.  and  ,n  a  dnecion  niakin«  an  anj-le  of  ,-•  4.  wth  the  1  .n- 
«ent  at  the  po.nt  of  ent.ance.  which  ,s  4  f..  .measured  ic.ri.  t  •  !  d 
-■  ft  .neasure,  verttcallv  fr,.,,,  the  sluice  whete  the  s.ream-l,„es  are'l.  r  - 
/ot.tal  hach  rub.c  foot  of  water  does  5^3*  f,..|hs.  of  useful  wJrk,,  r 
sec.  when  the  wheel  n.akes  43  revols.  per  „,,„.  Fin.l  the  f  ,T  ,  I  e 
wheel,  the  total  avadable  fall,  and  the  dia.n.  of  the  whe.  I. 

,  ,     .  .   ,  •'"*•  S.437  ft.,   10  ft..  24  ft. 

.-.  A  race  ,s  stratuht  and  close-fitfnK  so  that  the  lo.ss  of  e.fect  ,iue 
to  CM  ..pe  of  water  n,ay  be  disregarded.  .\  single  undershot  wheel  v.tl, 
p  ane  floats  ,s  replace,!  bv  four  s.milar  tandem  wheel.s.  If  the  dcl.ve.v 
of  each  of  the  four  wheel*  i.  the  same,  and  >f  .t  ,s  assumed  that  the 
water  reaches  each  w-heel  w„h  the  satne  velocity  with  whi.  h  ,t  leaves  the 
preced.HK  wheel,  ftnd  the  total  n,a.xitnu>n  dehveiy  due  to  impact. 

.       ^.  ,  ■■''"■  '1  ^>"'^*  tl'*-  'I'-livcry  of  the  single  wheel. 

3-  n.H."ss  the  preceding  example,  assuming  that  the  dehvcrv  of 
each  wheel  „  not  the  san,e.  but  that  the  total  .lehvcry  .,  ..  maxin.um 

■''"•  '  6times  the.lejivervof  (he  single  wh.el 
14.  If  «  wheeLs  of  the  same  type  are  substituted  for  the  single  wluvl 
...example  ...  an.l  ,f  „.e  assumptions  a,e  the  same  as  those  .n  example 
.3.  show  that  the  total  d.-l.ve.y  ol  ,lie  «  wheels  is  ,0  ,|,e  ,lel.ver^.  of  He 
single  wheel  in  the  ratio  of  2.  to  2,.  .  ,  and  that,  theoret.ca.ly.  ,f  the 
number  is  made  very  la.ge,  ihev  will  approximately  give  the  entire  wo.k 


A 


W^' 


F.XAMPI.FS. 


475 


15.  In  a  low  breast-wheel  (if  2.0  II.  (li.inieter.  the  water  enters  the 
hiirket  witli  :i  vcImciiv  of  16  ft.  per  .seCDiid  in  a  ilircclion  iijaking  angles 
>•{  45°  with  the  hoiiziintal  and  15°  wilh  the  wheel's  periphery.  The 
wheel  makes  7  revohitions  per  niiniite  ami  receives  5  cu.  ft.  per  second 
i)(  water.  Find  the  inechanical  etTect  of  the  wheel  and  tiie  positlun  of 
thr  sUiice,   which  is  placed  where  the  stream-lines  are  horizontal. 

A>is.  2075  ft.-ll)s.;  AD  =  2  ft.,  /i/>  :r-  .2;  ft. 

16.  The  water  in  .1  head-race  stands  4.66  ft.  ahove  the  sole  and  leaves 
the  race  under  a  j;.ite  which  is  r.iised  6  ins.  above  th"  sole,  the  coclTii  lent 
of  velocity  (r'l)  beiti^  .9;.  Tlie  water  enters  a  breast-wheel  in  a  direction 
making  an  ansjle  of  30°  with  the  tanjjent  to  the  wheel's  periphery  at  the 
point  of  entrance.  The  ^-peed  (//)  01  the  periphery  is  10  ft.  per  second, 
the  breadth  of  the  wheel  is  5  ft.,  the  depth  of  the  water  in  the  flume  is 
S  ins., and  the  leiif.;th  of  the  flume  is 8.2  (t.  Find  the  loss  of  head  (./)  due 
to  the  destruction  of  the  relative  velocity  (f)  at  entrance:  (/■)  due  to  the 
VI  locilv  of  flow  in  the  l.iil-race  ;  (c)  in  the  circular  Hume.     1/  —  .oiS.) 

.hts.  (,M  1.1  I  ft.;   ifi\  1.57  ft.;  iC)  .44  ft. 

17.  In  the  pieccdint;  example,  tiiiii  how  the  los.ses  of  head  would  be 
mollified  ii'  the  lliiiiie  were  lowered  1.03  ft.,  and  if  the  point  of  entrance 
wi'rc  raised  so  as  to  m.ike  «  ==  "'1  cos  30°. 

Alls.  (,i)  .939  ft.;  [6)  3.816  ft.;  tc)  t^d  ii. 

18.  20  cu,  ft.  of  water  per  second  enter  a  breast-wheel  of  32  ft.  di.im- 
eter  and  having  a  peiiphcr.il  velocity  of  8  ft.  per  second,  at  an  aiifjle  of 
25J°  with  the  circumference.  The  depth  of  tile  crown  is  1  \  ft.;  the  buck- 
ets are  lialf-fillcd,  and  the  fall  is  9  ft.  The  velocity  of  the  entering; 
water  is  12  It.  per  second.  The  centre  of  the  sluice-openinjj  is  .54  ft. 
above  the  |)oint  of  entrance,  and  the  width  of  the  sluice  is  3}  ft.  The 
wheel  has  48  buckets.  The  distance  between  the  wheel  and  breast  is  .', 
inch.  The  bucket  passes  tliroU).;li  .9  ft.  while  receiving;  water,  and  the 
deptli  of  the  water-surf.u'e  in  tlic  bucket  below  the  point  of  entrance  is 
1.25  ft.  Find  (.1)  the  aii(,'ular  distance  of  the  point  of  entrance  friini  the 
horizontal ;  {/>t  the  fall  in  tlic  breast ;  (c)  the  head  of  water  over  the  sluice; 
(</)  the  velocity  of  the  water  in  the  bucket  the  moment  entrance  ceases; 
(<•)  the  tot.il  mechanical  cfTect,  ilisrenardiiij;  axle-friction, 

//«i.  (.1)  53°  53':  <fi:(>.S-S  ft:f-»  \.9Mh.:{ifi  r4.9ft.;(c)  15.59  H.  P. 

19.  In  the  preceditii;  ()uestion,  if  the  enerj;v  absorbed  by  axle-friction, 
etc.,  is  743  ft. -lbs,,  find  the  etTlcicncy  of  the  wheel.  .Ins.   f_. 

20.  15  cu.  ft.  <»f  water  per  second  wilh  a  fall  of  8J  ft,  are  liroii^ht  on 
a  breast-wheel  revolving  with  a  linear  velocity  of  5  ft.;  depth  of  slirotid- 
in^  =  12  in.;  the  buckets  are  h,df-rilled,  and  ?'i  —  2/< ;  also  r,  =  12  ft. 
Find  the  theoretical  nicchunlcal  elTei  t.  y  being  30*.     Ans.  7040  ft.-lbn, 

21.  A  wheel  is  to  he  constructed  (or  ■!  jo-ft,  fall  having  an  8-it,  veloc- 
ity at  rircuiuferencc  and  taking  011  the  w.iter  at  12*  from  the  summit 
with  a  velocity  of  16  ft.  Determine  the  radius  of  the  wheel  and  the 
number  of  revolutions,  »'i  lieinR  2m.  Am.  12.9  ft.;  5.9. 


47() 


l.XAMrU.S. 


■•^> 


fc4 


2:.  If  foi  the  vvli<-cl  in  example  :i  llic  nunijcr  of  revolutions  is  5. 
and  V,  =  2«.  the  water  beinj;  a<;.iin  taken  on  at  \i\  tiri(i  the  radius  and  «. 

Ans.  13. 3r,  ft.;  7.1  ft.  per  second. 

23.  A  lirea^t-wlicLl  p.isses  12  cu.  ft.  of  water  per  second,  and  for  the 
speed  u  =  \v,  =  4  ft.  per  second,  the  loss  of  merlianical  elTcct,  due  to  tlie 
nl.itive  velo(  ity  /'  being  destroyed,  is  a   muiiiiinm.     Find  this  eflect, 

^  ^'"'"-  30'-.  ./;,,.    73.2  ft.-lbs. 

24.  In  a  breast-wheel  (J  ~  \o  cu.  It.  per  second  ,  //  ^  10  ft.;  -',  ^ 
;«  ;  «  =;  4^  ft.  per  second  \  y  -  lo" ;  diameter  of  >;;ndjri..on  ^  0  in.s.;  diam- 
eter of  wlieol  =  30  ft.;  /,  ^  .08;  weight  .)f  wheel  and  water  ^  20.000 
lbs.  I'"ind  the  niechaaical  effect  of  the  wlicel.  ( N'egh'ct  loss  of  effect 
due  to  escape  of  watei  fro  •.  buckets  and  to  frictional  resistance  along 
'''^' ""■.'••>  Ans.  5776  ft.-lbs. 

23.  The  quantity  of  uu.   r  laid  on  a  breast-wheel  by  an  overfall  sluice 
(>  cu.  ft.  per  second,  the  total  fall  being  S  ft.,  and  the  velocity  of  the 
periphery  5  ft.  per  second  ;  also  5;',   -:  S//.  and   if  ,/  be  the  depth  of  the 
shrouding  iMn  =  J^" (in  the  present  case  ,1'  -  12     is.).    Kind  the  elleitive 
(ail,  the  height  of  the  lip  of  the  guide,  the  angle  of  lU'  Iination  at  the  end 
i>(  the  giiiile-curve.  the  breadth  of  the  lip  of  the  guide-curve,  and  the 
radius  of  the  wheel  that  the  w.iter  niav  enter  langentially.     If  the  radius 
is  limited  to  12  ft    6  ins.,  find  the  deviation  of  the  direction  of  motion  of 
the  water  from  th.it  of  the  wheel  at  the  point  of  entrance.  ■  being  .f>. 
Alls.  0.9  ft.;  .325  ft.;  34"  46  ;   23  ft.;  3.S.6  ft.;   28'  3^.  . 
2(,     10  cu.  ft.   of  water  per  second  are  delivered   to  a  breast- wheel. 
The  total  fall  is  10  ft.     Th-  i)eripheral  velocity  of  the  wheel   is  6  ft.  per 
secon.l.     If  J-,  -  2«aiid  ;'  ^  30',  find  the  theoretical  useful  effect  and 
the  theoretic  al  <!rici<'n(v. 

A>is.  53584375  ft-lhs,;  .85735. 

27.  24  <  11.  ft.  of  water  enter  the  bin  kets  of  a  36-ft.  breast-wheel,  the 
total  fall  being  iij  ft.  ,\t  ih,-  point  .f  entrance  the  diiecti.m  of  the 
w.iter  makes  an  angle  of  30°  with  the  periphery  and  also  2v,  =  5;/.  I'ind 
the  mechanical  efted  of  the  wliecl  and  the  position  of  the  lip  ..f  ihe 
sluice  through  which  the  water  passes  to  the  wheel. 

.Also,  if  the  depth  of  the  shrouding  is  1  ft.  and  the  buckets  are  .uly 
half-tilled,  find  the  width  of  the  wheel. 

The  axle-bearings  are  6  ins.  in  diameter.  Taking  the  coct»i(  lettt  of 
friction  to  be  .008.  how  much  p<uver  is  absorl)ed  by  frictional  resistance, 
asbuiiiing  the  weight  of  the  wheel  and  contents  to  Ik-  ya.ooo  lbs..' 

Ahs.  26.83;  H.l'.:   .,    „-    .|f,i4ft.;  j<=:.5r,I5ft.;    ,o  ft.;  4.19  H  ( 

jS.  In  an  overshot  wheel  >,  -  15  ft..  1/  =  10  in..  $  =  \i>.  If  the 
•livisi.Mi  circle  is  at  one  half  of  the  depth  of  Ihe  crown,  liiul  the  angle 
(rii  between  the  bucket-lip  and  the  wheel's  ftcriphery.    (Take  .V  -  5»,.) 

./;/.    ri  =  i«*  I'. 

i').  An  overshot  wheel,  in  which  ,  ,  iS  ft.,  makes  4  revolutions  per 
minute,  ami  the  velo(  ity  ,,f  ii,c  w.itci  on  entering  tlic  bucket*  is  twice 


i 


l-X-IMPIES. 


An 


that  (if  tile  vvhr'<  1'';  i.cnphiTv.    If  ;-,  ^  :o°,  lind  y .  anil  also  t'lml  tlir  rela- 
tive v<-Ii'(  i!\   I  Ci  lif  tin-  (■iit<  riHL;  water. 

.///I.    1'  =  lo"  9'  ;    /'  =  7.7K  ft.  ]>i"i  sec'inii. 

-,c  If  Diic  fuunh  of  llic  tlieorutii  (•a|)arity  of  a  bui  kct  is  tiUeil  by  llie 
w.i'i:.  liiui  tlie  j;rcalest  luimljcr  of  l)ii(  kcts  thcori'ticaily  possitjie.  tin. 
(li'plli  of  tlie  irowri  biini;  I  ft.,  tlic  laflitis  (r,)  to  tin-  outer  |>eri|>li(ry  \2 
ft.,  ti:e  an^le  y\  m°,  and  tlie  velocity  of  the  entering  water  twice  that  ol 
the  wliecl's  periphery. 

Ans.   lovi-      Matcinp;  allowance  for  exit  of  air,  tlie  iiiunbcr  ol 
buckets  niij^hl  be  about  two  thirds  of  this  amount,  or,  say,  69. 

31.  A  wind  of  30  ft.  dianieler  with  73  buckets  iiiaki'S  7  revolutions 
iier  ininute,  (J  bein;.;  5  cu.  ft.  per  second.  The  division  circle  is  half  way 
between  the  outer  and  inner  peri|)heries.  If  </  -  I  ft.  and  7'.  =:  ://.  find 
the  efTect  due  to  impact.  .Ins.  514  ft.-lbs. 

3:;.  A  30-ft.  wheel  weighs  24,000  lbs.  and  makes  6  revolutions  per 
minute  ;  its  f;ud!.;e()nsare  6  ins.  in  diameter  and  thecoeiricient  of  friction 
IS  .o<S.  The  w.itcr  enters  the  wheel  with  a  velocity  c,f  13  ft.  per  second, 
and  in  a  direction  makiiif;  an  an^le  of  10"  with  the  direction  of  motion 
of  the  wheel  ai  the  point  of  entranie.  The  <ievi.ition  from  the  summit 
of  the  point  of  entrance  is  12°,  of  the  point  where  spilling  bcfjins  is  1  50  . 
of  the  point  where  all  is  spilt  is  160',  and  5  cu.  ft.  of  w.iter  enter  tin- 
wheel  per  second,  of  which  the  partially  filled  buckets  contain  one  hall. 
Uitermine  the  total  mechanical  effect.  .-//w.  ')ll4  ft.-lbs. 

33.  The  veloc-ity  of  the  oiiti-r  peiiplirry  is  <jfi  ft.;  ihe  an^le  between 
the  directions  of  motion  of  stieam  and  wheel  is  15°.  I"iiid  the  impulsive 
effect  of  th»-  water,  t,  In-inj,'  15  ft.  per  second. 

./«(.  (;i  ft.-lbs.  per  cu.  ft.  of  water. 

34  An  overshot  wheel  40  ft.  in  diameter  makes  4  revolutions  per 
minute  and  passes  300  cu.  ft.  of  water  |)cr  minute.  If  the  niidj;eons  are 
fiiiis.  in  diameter  and  the  wheel  weighs  30,000  lbs.,  by  how  miuh  will  the 
mc(  lianical  elTeet  be  diminishetl  ?     (/  =  .008.1 

.Ills.  25  ft.-lbs.  per  second. 

35.  The  diameter  uf  an  ovcrsho:  whet  1  =  30  ft.;  J'l  =  15  ft.,  //  =  rj 
ft  deviation  of  impinging  water  from  tliiection  of  motion  of  wheel 
iy)  r=  8J° ;  <leviation  of  point  of  entrance  Irom  summit  =12';  deviation 
of  point  where  spilling  begins  from  the  centre  =  5«J*;  deviation  <if  point 
where  spdiing  ends  =  70;*  :  fj  =  i  '"•  '*•  ^^^'^  '"'■''  cffwt  "'  impact 
and  weight.  -'"J.  if>.9  "  '*• 

36.  An  overshot  wheel  with  a  radius  of  15  ft.  and  a  12-in  crown 
t.ikcs  10  <  II.  ft.  of  water  per  se<-ond  and  m.ikes  5  revoaiiions  per  minute. 
II  w        1.  find  the  width  of  the  wlieel  and  the  number  of  the  buckets. 

.Ins.  5,',  It..  75  or  90, 

37.  An  overshot  wheel  of  31  ft.  diameter  makes  5  revolutions  j)er 
niiiiiiie.  Find  the  angle  iM-iweeii  tin-  water-surface  m  a  bucket  and  the 
hori/unta!  when  the  lip  is  140*  fnm  the  S'mimit.  .Ins.  4*  33'. 


478 


HXAMPUiS. 


*> 


iite 


4'.    * 

velocity 


3S.  An  overshot  wheel  of  lo  ft.  rli.im.-ter  makes  20  revohnions  per 
nnnute  I-,,,.!  th.-  anijle  betwee.,  H,,-  w.a.r-s„rla<:o  an.l  the  horizontal 
wDrn  the  lip  IS  (11  vo'  irom  the  sumin,!.  (2)  45-  26'  from  the  summit. 

,,.,  -•'"■>'•  '■'  .^4°  -V':  (2)  43'  iS'. 

39.  1  i'e  w.iter  enrers  an  overshot  wheel  at  12°  fiom  the  summit  with 
a  velocity  ..f  16  ft.  per  second  and  the  lin,-ar  velocity  of  the  wheels  ,.,- 
..phery  i.s  S  ft.  per  second.  The  fall  is  30  ft.  Find  the  diameter  ui  the 
wheel  and  the  number  of  revolutions  [ler  mimite.        h,^   -5  4  ft     -  05 

40.  In  a  r-h.  wheel  with  a  12-ii,.  crown  an<l  a  pripheral  velocity  of 
h  ft.  per  se.  o„d.  the  point  where  spdiinij  commences  is  doCmed  bv 
'I  =  0.  I-ind  thearc  .,v.r  which  >pillinj,  takes  place,  the  an«Ie  between 
the  l.uckel-arm  and  the  circumference  bein-  30°.  ,\lso  find  the  bu.  ket- 
allele.  If  I .  cu.  ft.  of  water  enter  the  wheel  at  15'  fiom  the  summit  with 
a  veh.cityof  18  ft.  per  second,  hnd  the  mechanical  effect  due  to  im- 
pulse and  to  weiyli',  l-  bcinj{  i. 

Ahs.  arc  ^  15.2  it.;  (i  ^  56=  33';  2,87  II. I'..  jS  JJ.!' 
,vershot  wheel  of  32  ft.  diameter  levolvcs  with  an  anijular 
.ow  that  the  allele  between  the  horizontal  and  the  watei- 
surface     .  a  bu   ket  at  90'  fioin  the  summit  is  tan    '  " 

4:.  A  water-wheel  has  an  internal  diameter  of  4  ft.  and  an  .Morn  il 
diameter  of  8  ft.;  tlie  directi.m  of  tlie  enierinn  water  makes  an  ani,.le  of 
15'  with  the  tangent  to  the  circumference.  Kind  the  angle  subten.lcd 
at  the  centre  ol  the  wheel  hv  the  buck.  t.  which  is  in  the  form  of  a  circ.i- 
lar  arc,  and  also  tind  the  radius  ol  llie  bucket.     Aiis.  28"  54';   1  -,->■',  ft 

4J.  An  overshot  wheel  5  ft.  wide.  30  ft.  in  diameter,  h.ivin«  a  ,  ..jn 
crown  .m<i  72  buckets,  receives  10  cu.  ft.  of  water  per  second  and  ni'ikes 
5  revolutions  per  minute.  Determine  the  deviation  from  the  horizont  il 
at  which  the  water  begins  to  spill,  an.l  also  the  correspondi.,.'  denres- 
si..n  ol  the  water-surfa.e.  Jns.  31 "  41'  ;'"'5'  5,'.  ' 

44.  An  overshot  whe.l  makes  ^^  revolutions  per  mmute  .  its  meai. 
ftiameier  is  32  ft.;  the  water  enters  the  buckets  with  a  velocity  of  8  ft 
per  s.-cund  at  a  point  i:'  30'  dom  the  summit  of  the  wheel.  \t  the 
p.unt  of  entrance  the  p.ith  of  the  mflowins,'  water  makes  an  ancde  of  to" 
with  the  horizontal.  Sh..w  that  the  path  is  horizontal  vertically  above 
the  centre.  The  sluice-board  is  placed  at  a  point  whoso  horizonl;i|  dis- 
tance from  the  centre  is  one  half  that  of  the  p.,iMt  of  entrance.  Find  its 
position  rel.itively  to  the  centre  an.l  its  inclination  to  the  horizon 
•^'*"  ""^   '■•  -''"•  I^'  6' ;  6.24  ft.  per  ,econrJ.    ' 

45.  Ihe  water  enters  the  buckets  of  the  wheel  in  the  precc.iini' 
example  without  slicurk.  Find  the  clbow-anKle.  Also,  if  the  buckets 
be^in  to  spill  at  150*  from  the  summit,  tin.l  where  the  bucket  is  empty 
and  the  number  of  buckets.  (Depth  of  crown  =  .2.  ins  ■  ihickn  as  of 
bucket  =.  i\  ins.)  ^,„    ,,3-  3„,.  ,j^,  ^^^ ^ 


nfir 


EX.tMPIHS. 


479 


46.  Given  -/,  =  15  ft.  per  secon  !,  aii.l  0  -r  2',\\  F-iiul  tlio  position 
of  the  centre  of  the  sliiire,  which  is  4  ins.  above  tlie  point  of  enir.incc. 

.;«t.  .0877  ft.  vertically  below  and  r.0143  ft-  horizontally  from 
the  summit.  The  a.xis  of  the  sluice  is  inclined  at  </  33'  to  the 
horizontal.     (Assume  y  —o'.) 

47.  In  an  overshot  water-wheel  v,  =  15  ft.;  „  =  10  ft.;  elbow-anKle 
7ol^    division. anj^lc  =  4^;  deviation   from   summit  of  point    of    cn- 

tr.mce  =  12'.  Find  the  deviation  of  the  layer  from  th.it  of  the  arm,  so 
liiat  the  water  mi^lil  enter  uiiiinpedod  ;  also  find  the  mrhnalion  of  the 
Kiyertothe  horizon,  and  the  value  ..f  / '.  If  ilie  centre  of  the  .sluice- 
aperture  is  to  be  4  ins.  above  point  of  entram  e.  lind  its  verti<  ul  and 
liorizontal  distance  fioin  the  vertex  ol  the  streams  parabolic  path  which 
is  vertically  above  the  centre  of  the  wheel,  and  a!>o  tin<l  inc  lination  of 
shiice-bo.ird  to  horizon. 

Im.    iV;  ioy  ;  5.3  ft.  per  second  ;  .0S78  ft.;   1.04  ft.;  9"  34'. 

4S.  A  wheel  makes  jo  revolutions  p.r  minute  ;  tadiu.s  —  5  ft.,  anfjle 
of  disch.irg.  =  o".  F'ind  deviation  of  water-surf.ice  from  horizon.  .Mso 
find  ricviation  at  44"  35'  .-.bove  centre.  ../«>.  4'  3;' ;  4^^  ,e,' 

40  In  an  overshot  wheel  <2  ^  18  cu.  ft.  ; /■,  =6  ft.  ;  ,i  =  I'ft.;  iJ:=4ft.: 
,V  =  ::4  .  H  -  17.  At  the  moment  st>illini,'  commences  the  area  i/^/,/  .= 
1.025  s'l-  ft-;  between  this  point  and  the  point  ulure  the  spilling  is  om- 
pieted  three  buckets  are  interposed,  tlie  sectional  areas  of  the  w.iter 
beiiiK  .501,  .409.  ami  .195  sq.  ft..  res|)ectivL-ly.  Find  i,i)  the  sectional 
area  of  bucket ;  {f>)  the  p<jint  where  the  spillinf;  commences;  (t)  the  point 
where  the  spillin},'  is  com|ileteil  ;  (,/)  the  hei;;ht  of  the  arc  of  discharge; 
U)  the  mechanical  effect  due  to  the  fall  of  the  water  through  the  arc  t.l' 
disch.iri;e,  y  bciufj  10'  46'. 

Alls.  (,i)  .662  sq.  ft.  ;  i/'i  0  =  7"  i  3',  0   -  :S'  46'  ;  (<)  O  :=  73'  -.  ■,' 
«'*=5'5'':  f'/)4-49ft.;  (.•)4.y,5  H-I'. 

50.  In  the  preceilin;;  exatnple,  if  the  w.tlcr  enters  with  a  velocity  ,,f 
20  tt.  per  second  at  2o'  below  the  summit,  and  if  the  direction  of  the 
inflfjwinK  stream  makes  an  an),de  of  23' with  the  wheels  periphery  at 
the  |>oiiit  of  entrance,  find  the  mechanical  effect  1,1)  due  to  impulse; 
(/■I  due  to  the  fall  to  the  point  where  spiliini;  commences. 

.his.   (.1)  5  08  HP.  ;  (/;,    12.114  n.P. 

51.  300  cu.  fi.  of  w.iter  per  minute  enur  the  buckets  of  a  40-ft.  over- 
shot wheel  with  a  i2-in.(rown  and  making  four  revohuions  per  minute. 
The  wheel  has  136  backets.  Ai  the  moment  when  spillinR  commences 
tlic  area  K;// -  \zf.i  sq.  in.  The  spilling  is  complete.l  when  the  anfrle 
between  the  horizontal  and  the  ra<liu»  to  the  lip  of  the  bucket  ^  62*30'. 
Iktween  these  two  positions  three  buckets  are  interposed,  the  sectional 
areas  of  the  water  in  the  b.u  kets  beini..  24.5,  14.4S.  and  6.6 -sq.  ins.,  respec- 
tively. The  veitical  distance  between  the  water-surface  in  the  first 
bucket  and  the  centre  is  18  ft.  Find  (,»)  the  wi<lth  of  the  wheel ;  (/>)  the 
cross-section  of  a  bucket;  1.)  the  angle  between  (he  horizontal  an<l  the 


48o 


FX.IMPLFS. 


f- 


sin  '>,.  r  b.-in^'  the  (lisi.iiicc  (a^-iumeci  (.-oiistant)  .,f  the 

liic   liiicir  veluc  ity  of  tlic 


r.i.lius  t„  the  lip  .,f  thf  l,u.  k.t  wh.-ii  spilli„<.  rommences  ;  (,/,  the  hci-ht 
of  the  disc'uir{;in,<r  an  ;  (,'i  ili,-  hum  h.inKal  cllc.  t  due  to  \vei<;ht 

Ans.   it)   J.4  ft.  ,  i/l  3,,2S  .,,.   fi.  ,  ,,  ,  0  _-  52'   i,;  "  ,,/,   ,  ,,  ft   . 

f)  «9-73  I  .1'. 

5:.  As    the   biK-ket-.inn   <,/  ni..ves   (lownwanl    fn„„    thr    '■onzontal 

position,  show  that  while  the  wlieel  moves  through  an  aiij-le  0  the  last 

particle  of  water  at  c  will  move  through  a  distance  approximately  eoual 

to '1^': +-«■'>  ,0 
ir 

particle  of  water   from    the   axi■^,  .mil    //   bein 

wheel  at  the  radius. 

53.  If  the  last  particle  ..f  uat<T  leaves  the  buckets  just  as  the  lip  d 
re.iches  the  lowest  point  of  the  wheel,  and  if  the  artn  is  I  ft  iti  leni'th 
liml  the  anpile  between  the  lip  and  the  wheel's  periphery  (i  ,  l,,r  a  wheel 
of  3oft.  diameter,  the  peripheral  velocity  being  ;  ft.  per  s.,,,„,i"  (,,  ,„, 
a  wheel  -f  40  ft.  diameter,  the  peripheral  velocilv  being  10  ft.  per  se.'ond  • 
(3)  for  a  wheel  of  10  ft.  diameter,  the  peri[)heral  velocitv  bein-  S  ft.  per 
second.  ./,,,   /,.  ,„■>  .  , ,-,  ,     .>"  , 

54.  In  an  overshot  «he.>l  of  30  ft.  diameter,  5  cu.  ft.  of  water  per 
second  enter  the  buckets  with  a  vel.,c,ty  of  16  ft.  per  second  and  the 
wheel's  velocity  at  the  division  <  ircle  is  7  ft.  per  second.  The  point  of 
entrance  is  iS"  from  the  summit,  and  the  angle  between  the  directions 
of  the  inflo  ving  water  an<l  the  wheel's  periphery  at  the  point. ,f  .■nir,Mce 
IS  iz\  The  water  begins  to  spill  at  M.sr  fr.,m  the  summit  and"  the 
spilling  IS  complete  at  160^  from  the  summit.  r,nd  the  t..t,,l  mech  ui- 
ical  elfect  due  to  Impulse  and  weight.  What  is  the  t.uit;,ntial  force  it 
the  outer  periphery.'  ./„,     k,..SII1'.     1194  1I,s" 

35.  In  a  y.-h.  whe-  1.  with  a  ift.  crown  and  .i  p-r.pi,e;.i'l  veio,  i,y  of 
8  ft.  per  second,  the  point  where  spilling  commences  is  defined  by  the 
relation  0  =  0.  Fi.id  the  arc  over  which  spilling  takes  place,  the  angle 
between  the  arm  and  circumference  being  30°.  Also  lind  the  •  l,i„  ket  ' 
angle.  If  11  cu.  ft.  of  water  enter  the  wheel  at  i;  ins,  fmm  the  s,i„„„„ 
and  with  a  velocity  of  18  ft.  per  second,  show  how  to  lin.l  the  meclum,  .d' 
ellecl  due  10  impulse  and  that  due  to  weight. 

A>u.    53°  23'  ;  4-  ,   ,r,,2y  ft.-lhs.  ;   14.406.56  fi.-lbs  per  second 

56.  An  overshot  wheel  of  32  ft.  diameter  makes  y,.'  revolutions  per 
mmiite.    Find  the  mclin.ition  to  the  horizontal  of  the  v  .itei-surf.ice  in   i 
bucket  at  90°  from  the  summit.      If  the  wheel  has  <;o  bii.  kets  an.l   wi 
arms  make  an  angle  of  22'^  »itli  ilie  periphery,  lind  the  ,1,  ,,ih  .  (   th. 
^^rown.  ^|^^    _,  jj,     ^^  ^ 

57.  An  overshot  wheel  of  3;  ft.  diameter  makes  ',"/  n'Volutions  per 
minute.  Fm<i  the  inclination  to  the  horizontal  of  the  waler-surf.ice  iii  ■, 
bucket  at  90- from  the  summit.  If  the  wheel  has  go  buckets  and  the  irms 
make  an  angle  of  22l°  with  the  periphery,  find  the  depth  of  the  crown. 

Ans.  25.68  ft.  ;  5  04. 


I-,. I 


/:X.IM/'ltS. 


481 


58.  All  ovcrsliot  vvhool  of  36  ft.  diaincter  and  witii  9(1  htitkets  h.i-,  a 
peripheral  velocity  of  -!,  ft.  per  sccon-1.  Tlio  watcrcnt.rs  wnii  a  veio,  ity 
of  15  ft.  per  second  and  acquires  in  the  wheel  a  velocity  of  16.49  ft.  per 
second.  Find  the  distance  throned;  wiiich  the  float  moves  during' 
""I""-  _  .//;.,.  r.  15  ft.  " 

5';.  1  he  sluice  f.ir  a  10-ft.  oversli<jt  wheel  is  vertically  above  the  cen- 
tre and  inclined  at  45-  to  the  vertical.  Thf  w.itcr  enters  the  buckets  at 
a  point  2  ft.  vertically  helow  the  sluice  and  10'  from  'he  summit  of  the 
wheel,  I'ind  the  ans;le  between  the  directions  of  motion  of  the  enterini; 
water  and  of  the  wluel's  circumference.  .\lso  find  the  velocity  of  the 
water  as  it  enters  ih.>  wheel.  .Int.  5°  30' ;  9.68  ft.  per  secorul. 

60.  In  an  overshot  wheel  v,  -  17  ft,;  //  =  11  it.  per  seiond  ;  elbow- 
aii,i;le  =  70°;  divisiiiii-ani-le  -  5°;  water  enl.-rs  the  hrst  bucket  at  12' 
irom  summit  of  wheel.  Find  (,j)  the  idative  velocity  /'so  that  water 
may  enter  unimpeded  ;  i/j  the  direction  of  the  entering  water;  ic)  the 
<liameter  of  the  wheel,  which  makes  5  revolutions  per  minute;  (,/j  the 
Iiosition  and  direction  of  the  sluice,  which  is  2  ft,  measured  horizoiit.illy 
from  the  point  of  entrance. 

.his.  6.24  ft.  per  second  ;  ^     =  7'  13';  42  ft.;  45  ft.;   5' 43'. 

61.  In  ail  overshot  wlieel  the  <ieviation  of  the  impinging  w.iter  from 
the  direction  of  .notion  of  the  wheel  is  10'  ;  the  velocity  (?■,)  of  the  im- 
pinging stream  -  15  ft,  per  second  ;  of  the  circumference  of  the  wheel 
(//(  =   1  :;  cos  10.      What  amount  of  the  heail  is  sacrificed  ^ 

.his.   1.06  ft. 

62.  .\  30-tt.  water-wheel  wiih  7:  buckets  .ind  .1  12-in.  shrouding 
makes  5  revolutions  and  receives  240  cu.  ft.  of  water  per  minute.  Find 
the  width  and  s<-cti  )nal  area  of  a  bucket.  The  fall  is  30  ft.;  at  what  point 
does  the  water  enter  the  wheel,  the  mtlovving  velocity  being  il  tunes 
that  of  the  wiieei's  periphery.^  .Mso  tind  the  devi.ition  of  the  water- 
surface  from  the  Iwirizontal  at  the  point  at  which  disehar';iiig  com- 
mences, i,e,,   140'  from  the  siiiiimit. 

.//;>.  2,03  ;  .327  sq.ft.;  32'  47';  4-  iS'. 

63.  What  number  of  buckets  should  be  given  to  an  overshot  wlieel  of 
40  ft.  diameter  and  12  ins.  width  in  wheel,  pitch-angle  -  4  ,  ihickness  of 
bucket-lip  =  1  in.,  water  area  -  24i  S(|.  ins,.' 

.Ins.  167,  depth  of  crcjwn  bting  9  ins. 


(   11  \i'  I  I.R     \'II. 


If 


'1^ 


■ni<i!i\i:s. 

I.  Reaction  and  Impulse  Turbines.  -All  turMiics  hd.. ri- 
te >    i<nc    (){  tuo  cl.issc-^,    \i/.,    Kiiutti'ii     I iirbiihs    ami    /ii/f/'';,- 

'/iirhiiiis.  and  arc  ile.sij,nic(l  tn  utilize  nir-rc  (>•-  less  of  the  a\ail- 
;ihle  encri,')'  of  a  iiiox'in;;  mass  of  water. 

ill  a  n;irt/i>ii  turbine  a  jx.rtinn  (.f  the  axaiLiMe  encr'.,'v  i-; 
C()n\erteil  into  kin.  tic  eneri;_\-  at  tlie  inlet  surCace  of  the  wlieel. 

1  lie  water  enters  the  wheel  jiassaires  fonneii  by  suitably  curved 
\anes.  and  acts  upon  tlicse  vanes  by  pressure,  eausinf  t!ie 
MJieel  to  rotate.  llu  jM-oportiniis  ,,f  the  turbine  are  sueh  tlmr 
there  is  a  particular  pressure  heme  the  term  ])ressure-turbine 
at  the  inlet  surface  eorrespondin.L;  to  the  best  normal  condition 
of  unrkini;.  .\n\  variation  from  tin's  pre-snre.  caused,  e.".. 
I)\  the  ])artial  closure  of  the  i)assaLres  throuLjh  which  the  water 
passes  to  the  wlieel.  Jian-e-  the  workin-;  conditions  and 
dinnnishes  the  eflicienc\-.  in  order  to  a\(iid  such  a  \ariatioi) 
of  jiressure.  it  is  essential  that  there  should  be  a  contimiit>-  of 
flow  in  everv  part  "the  turbine:  the  wheel  passaijes  sjiould  be 
kept  conipletel)  filled  with  water,  and  therefore  must  rt-cei\r 
the  water  slmultaneouslx  .  .Such  turbines  .-re  said  to  ha\e 
complete  adnn'ssion.  The  admission  is  D.irtial  when  the 
water  is  received  ox-r  .i  portion  of  ih<'  in  et  surface  onl\-. 

In  an  /w/'/^/.vc  (iirar<l  tin  bine.  I"ii,'s.  2~~  and  2-S.  the 
rnerLjy  of  the  water  is  wiiolly  converted  into  kinetic  cner'n-  at 
the  inlet  surface.  iluis  the  water  enters  the  wlicel  with  a 
velocity  due  to  the  total   available   head  and   therclore  without 


"•Iff' 


kli.-IL'llOS    .-IM)    IMin  LSI      ILRHISHS. 


iS; 


jircs-iirt-.  i-^  rrcci\c(l  upon  tlif  ciirwd  \aiU'-.  and  iiii[)art^  to  ti)e 
whci-l  tlu-  whiiK-  III  its  I'nrrL^y  l)y  nu-aii^  .i|  the  impulse  due  tc» 
tin;  .gradual  (.lianj;i-  <A  iiicniiriUuni.  ('arc  must  be  taken  to 
insure  thai  tlu  water  ni.i\  l)e  treel_\-  deviated  on  tile  eurxeii 
vanes,  and  heiue  sir  h  turhitu-.  are  sometimes  eallcd  turliines 
with  tree  deviation,      i-'orthis  reason  the  uater-passayes  sliould 


Kii:    277.  Fic.   i-S. 

r,ir.ir.l    liirl.iiie  f^r  I,i>w  Falls.  (iirard    I'luLiiu-  lo:    lli^h   l",ills. 

never  l)e  eomplctoly  filled,  and  the  \\,ner  should  tlou  tli|-oi:i,'ii 
under  a  j)re-~siire  uhieli  remains  eonstanl.  In  order  to  insure 
an  unbroken  llou  thron-h  the  ulieL'I-pass;i,L;es  aiui  that  no 
e  liht  -  are  tormed  at  the  baek--  of  the  \,iiu-s,  \  I'ntihitin:^-  Ilole.-^ 
arc  arran^'cd  in  the  wheel  sides,  I'it;.  jSo.  I'"ij;s.  2jt)  and  :rSe 
;ds(,  show  the  relative  ])ath  A/!  and  the  absolute  path  I'P 
tr.nersed  In  the  water  in  an  inward-flow  and  a  downward-flow 
tiirbme. 

It  there  is  a  sul'ti'  ient  he.ul,  the  wheel   may  be  ])iai  ed  elear 


'> 


^^■^  Rl--t<    ll'->\    .•/'./'    IMrLl.S/:      /7AV;/W:,S. 

.il.MV.     tl;c   t.ul-uatcr.    uiicn    lli,-    Mrcam    will    1„.    at   all    times 

unilrr  atiiiM-|.li,ri(.  prossur,-.  \\  ,(], 
1"U-  tall-  the  uhcM  1  ,„ay  la-  ,,i.„.,.,i 
in  a  (MMii-  -iipplnil  with  air  trnm 
an  air-punip  hy  wlikii  the  Mitlaic 
J;  "'  ^''^'  water  nia\-  he  kejjt  at   an    in- 

'-  .  variable  iewl  |,el.,u-  the  ..utlet 
orilkcs.  whieh  is  essential  fur  prr- 
fectl)-  free  ileviatimi.  While  the 
w  lieel-passa'^es  of  a  reaction  turbine 
should  !,.  kcpte..nipletel\t-,l!c<l  with  water,  n-.  such  restriction 
IS  necessary  uith  an  impulse  turi)ine,  Ti,e  supply  may  b- 
J'.irtiallN-   checked   and   the    water   may  he    received    by  one  or 


\     /    /'. 


279- 


TAIL  WATER 


Fic.  :So. 


N 


more  v.incs  without  atVectinL:  the  efficicncv.  Thus  t!,e  dimen- 
sions of  an  impulse  turbin.-  mas  varv  between  v,tv  wide  Imnts 
■■■'  "'"  '"'•  '"^;l'  tails  uith  a  small  suppK  a  coniparativelv  lar-e 
wheel  XMth  low  speed  m,,>  I,,-  empIoyed^  The  speed  of'a 
reaah.n  turbine  under  simil.ir  con, htions  would  be  disadvan- 
t..5ieously  -reat.  an.l  an\-  considerable  incr.Mse  of  the  diameter 
would  !ar,4ely  iiu  rea-e  the  tluid  t,  iction  and  would  also  render 
tl"'l'r"P'-  l"-l-ri,onm^  nl  the  vane-an^les  almost  impracti- 
-  ■il.le  Impulse  turbines  may  have  complet.^  or  partial  admis- 
■•o",  Ahile  in  reaction  turbines  the  admission  shouhi  he  alwavs 


W- '  ^1  ■ 


'IHt     Hi  Kl<)   (U'RUY   A\l>   l'hri<i\    UHSIJ. 


4><? 


tninpktc.  a-  ill  1\l;  jSi,  uhiJi  slmu.  the  r(!,iti\<  ],.ttli  .•//>' 
and  ahsnlute  jialli  (  /'  traM-r-^cHl  1)\-  the  watrr.  \\!un  then'  i^ 
an  ample  ^i:i)!i!>  <il  uatcr  the  rcactiur,  tiirhiiu  i-  iiMiallv  t<i  \k 
preferred,    l>iit    <:ii    \er>     In^li    tails   it-    specii    beceniic-    iiKoi, 


GUIDES 


veniently  i,'reat  ami  it  is  tli<  n  hettcr  to  aiHipt  a  turliir.r  -i  the 
impuNe  t>pe.  The  diameter  of  the  uluel  ean  then  he  iiu  leased 
and  the  --peed  i>roportionatel\-  iliminivheil. 

i  he  /fi(r,/_r-i;!ir,iv  i>  the  name  popularly  L:i\en  to  an 
mipulse  wheel  which  was  ijitiodueeil  into  the  miiiin-  district'-- 
of  California  about  the  \ear  iSl,;.  Aroimd  the  iXTipliery  of 
the  wheel  i-~  arrail-ed  a  series  (,f  t1,it  iron  hnekets,  .ihoiit  4  'o 
6  ins.  in  width,  whieli  are  stniek  nonii.illx  In  .1  jet  of  w.it.r 
often  not  more  th.m  three  ei-hths  ,,(  .in  nu  h  m  di.imeter, 
'1  heoretically  the  eftieieiu)  of  sueh  an  arrangement  ,:niii()t 
exceed  50  ])er  cent,  wlnle  in  jiractite  it  rarelv  reaches  40  per 
cent.  ihe  best  si)ced  of  the  wheel,  in  accordance  with  both 
theor\  and  jiractice.  is  about  one  haif  of  th.it  of  the  jet  .\lthouylv 
the  etlicieiu  \  i--  so  low,  the  wheel  found  .L;re.it  favor  lor  many 
reasons  .\n\  re(juired  speed  could  be  obtained  lu'  a  suitable 
choice  ol  diaineter;  the  plane  of  ihe  wheel  conl.l  be  i)laced  ii: 
an_\-  conv.'iiient  ))osition;  the  wheel  could  lie  choapU-  con- 
structed and  was  l.iri;elv  free  fr.-.m  habi!it\  to  ,m  ident  I  Icnrr 
it  was  of  the  utmost  im|)ort,inie  to  incn  .ise.  if  possible,  the 
cfficieni\-  "f  a  wlieil  possessm-    ,ueh    ad\,inta-es.       ObviousU- 


4^6        MCTi.u  r.'iiH  ')/  /uin  r.iKiK.ii   i\  tlkh'm.. 


•  * 


'te, 


'N, 


^A 


':J 


n  fir-^t  -^tf])  v.a-^  to  -^uhNtitutr  i.uj>^  tnr  tin-  flit  liuckft.-..  iW 
minu'tliati-  rr-^iilt  lU'tL^-^arilv  IniiiL;  a  \  ci)  lar-c  iiurrasc  m  the 
tfluifiic)-.  1  hi-  \\a>  incrca-K'd  -till  liiitlur  li\-  tlic  ailMjitinn  of 
double  buckets,  Im-.  jSj.  that  i-,  curseil  hikkct-  (.la  ale, 1  in 
the  iniildle  so  that  the  watei-  i.  e(|iial!y  detlceted  mi  Ixitli  saUs. 
I'hus  (!eveln])c(l,  the  u  lu(  1  i-  u  idely  and  m.,-,t  ra\orabl\- 
known  as  the  IVlt.in  wheel,  Iml;'.  :!Sj.  It.  ehieieiu\- is  ,it  lea-t 
.So  i)er  cent,  .iiid  it  i-  el, timed  tli,it  it  often  ris^s  .ibovo  <)0  per 
cent.  'I'lle  pow  ir  of  the  wheel  dm-,  n..t  dep.iid  tipnn  its 
tii.nnieter.  l)iit  ujion  the  ,i\  .lil.ihK-  iiuamit\  .md  liead  i.f  u.iter. 
The  water   i)a.--e>   to  the  wheel    throii-h  one  or   ni.iie  no//lcs, 


Fie    3^2. 

h.ivinq;  tips  hnreil  t.i  suit  ,ni\-  recinired  ,!c  Ii\ir\\  Idles,-  tips 
arc  screwed  into  the  no/zle<  ,ind  e.iii  he  e.isily  .uid  r,ipidl>- 
replaced  b\-  others  of  lar-er  or  smaller  -i/e.  -.>  that  the  Peiton 
is  espcciall\-  well  .ad,i])ted  for  a  v,ir\inL;  -uppl\-  of  u.iter.  It  is 
cl.iimed  that  in  tins  m.itiner  the  jiower  ma\'  be  wuied  from  ,i 
maximnm  down  tr.  j;  per  i  ent  of  the  same  w  ith<  ait  apnreci.ible 
loss  of  effuiiMUN  . 

2.  Actual  Path  of  a  Fluid  Particle  in  Passing  through 
a  Turbine.  — I'nder  the  comhmed  eftect  of  the  inlet  \elocit\- 
'\  and  tile  rotation  of  the  wheel,  .i  Ihiul  p.irtiele,  enteritii,'-  at  ,/, 
wll    traverse   ,in    aetii.d  p.ith  r?'\  uttin-  the  <iiit!et  iiir';;-:- at  an 


4. 


ACTi'.ti   riTH  Oh  111  in  r-iKiicii-  is     -kiush.         \^l 


f 


,in^U-   equal    \"  'V       1  lii--   path    may   he   appnixiniatcU-  plotted 
111  tile  t'dHduii:-   ill, inner; 

Lot    clf.     <)'/      he    twii    ^(.n-^eLiitivc 
})!a(ie^. 

I.et  '/  l)e  tile  (';i^eliarL,^e'  ]Hr  -ecnnil 
tlirciu^h    tile    pa^>a-e    hetwtv^n     these 

!)l,uies, 

1. -t  xx  lie  a  surfaee  concentric 
uitli  iKi'  and  lit  radius  /•. 

Let  /  hi-  the  lime  in  -eennd-^  in 
which  a  tluid.  jiaiticle  lluw-^  tr'nii  ,iii  tn 
i.i  . 

Let  .1  he  the  a;cM  iiix  <i  . 

Let  (/  he  the  mean  depth  ot  this 
.ircM  hetween  the  crovui-. 

Let  u>  he  the'  aiv^ular  \el<icity  of 
the  w  hei-l. 


Kir..  23;j. 


Ihen    (V  \.)lunie    , if  v\  aler  hetween  ^r-?    and.c.r  .hi. 

I'.iit  in    the    >ame  time  ,'  the    point  .r  \ull  have    movetl    to  -, 


X-  =  rui/  =.  n.ui   -. 
1 


In  tlii>  eiiu.iti.in  tl-.e  \ahies  nt  ui  and  ./  arc  known,  s,,  that 
by  de-crihiiii;  ,in\  ie.[iiirenl  number  ot"  c\-liiulrical  surfaces  and 
introiiucinL;  i"*  >  the  ei|uation  th<'  correspondinij;  v.ilues  ol  /  ,  ,/, 
and  J,  a  series  ot  v.dues  will  be  obt.iined  fur  i  .  detlnini,'  the 
points  -  ,  .?., ,  J,  .   .   .   on  the  ,ictu,il  p.ith  (f/of  the  tluul  p,irticles. 

/eiuier  skives  ,i  someuliat  nmrc  _^'oncral  method  as  follows: 

(.  oiisider  a  tluid  particle  moving  along  the  axis  AM/ of  the 
])assage  helueeii  two  consecntive  vanes  (?/"and  a  f . 

If  the  wheel  were  at  re-t,  the  jiarticle  in  t  seconds  wmiiil 
reach  ertain  |)oint  M.  but  the  rcnation  of  the  wheel  carries 
It  to  .1/   ,  where  MM    —  ru<t.  r  being  the  radius  (1)1  (—  O.U  ) 


48S  ACIUAL   PATH    Oh   llitll>   r^KTIUF   AV    TLKHINE. 

aiitl    a>  the    constant   an-ular    v<:lucit\-   al.oiit    tlic    axis   at   (>. 
'I'lu.s,  after  /  sccoruls,  M    is  tlu-  true  L.cus  ,,f  the  llui.]  i)articlc. 


Fid.   SS4. 


U^ 


I.ct  </»  and  <,V  Iw   tlie   angular   .loviations   <.f  i\][  an,!  Ol/' 
fr-ni  (M'. 

I.«'t  //,  (       .1/  /',  bo  tlic  linear  velocity  of  J/'. 

I-it   /•,  (        .?/7),  h,.  .1,..  r,  l.tixr  xrl.uity  ..ftlic  fliil.l  particle 


at  J/ 


Lt7',  (-  .l/'.\)  l.c  tlleal) 


e  absolute  velocity  of  the  lluid  particle 

at   .1/  .  ' 

let  H  !)<•  tlx'  anjjle  between  / ",  and  the  radius  OM  or  OM  . 

Let  H   l)e  the  anjjle  between  ,■,  and  OM' 

Then 


1 

t: 

and 


V,    .Hill    H'    .        !l  ,  l\     sin    H 


Vj,  cm  H'  ^   I  \  Cos  « 


•pr 


yicTUAi   r/iiH  or  hirii'  r/iRTicLh.  in  turbine.        489 


IlclK 


-rj^   '     ^=  tan  i< -\-  tan  W. 
V^  cos  f/  ' 


l?ut  //    =:  raj;   tan  f*  =  r   .  ;   tan  H   ^  ^  r    ,   ,  sinec   .1/  ,^    is 


necessarily  tanj^cntial  to  the  actual  patli  KM  at  .1/  ;  and 
./,/'^cos  ^  =  (J.  tlie  volume  of  flow  jkt  second,  yi ,  beint;  the 
sectional  area  of  the  passage  at  rij^ht  anyles  to  OM.  Substi- 
tutinj;  these  values  in  the  last  eiju-ition, 

M  </</»         </0' 

(T'  ~    Jr  +    ,ir  ' 


,uul  tliLref^ro 


a) 


./:'■ 


I,,// ..  <,'>+</>', 


I     beinLT  the  intern. il  railiu-.  <.<{  the  uheel. 

lUit   the   expression    /  A<ii    is    thcNdhinie   lA   tlu    passage 
between  tUt'  .ind  .1/ ami  ina\- i)e  deterinim-d  b)' actual  nicasure- 


nunt. 


Designating;  this  volume  b\    {',  ,  thei;, 


«u 


r,     t,^  f  0'. 


•in  ((luatioii  {{ivinp  (,'>  when  0'  is  known,  or  <p'  when  0is  known. 
I  liii .  the  actual  jMJsition  of  the  particle  can  Ik-  determined  if 
its  relative  jK)sition  is  known,  or  its  relative  position  can  he 
found  when  its  actual  j>osition  is  yiven. 


490 


Cl./iSSinCATH        Oh    TUK8INF.S. 


'lake  ii  mimhcr  i)rc(iui(list;int  pnints  .I/| .  .1/.,  .l/"  .  .,l,)n.r 
the  axis  of  tlic  piissa-^c,  aii.l  Id  (/), ,  0.,,  ,,')^  .  .  .  |„-  tin-  an^'Lil.u 
deviations  of  ().I/| .  OM,.  (>.]/,  .  .  .  from  ()A'. 

Also,  \ct  (  ^  .  (  ,.  f  ^  .  .  .  he  the  volumes  of  the  passage 
between  ,r,i  and  .I/,,  dii'  and  .}/".,  ,i,r  anil  .1/,  .  .  .  Tlien  the 
anj^ular  (ic\iations  ,/./,  0.;,  0.^'  .  .  .  ,,f  tlie  radii  to  the  corre- 
sponthn-  j„,ints  .I/,',  ^r.\  .?/,  .  .  .  on  tiie  actual  path,  are 
given  in-  tiie  eijiiations 


<.-'■ 

:';  +  ■.'','■ 

l'- 

i;  1-  •,%'. 

"r. 

<■>  -4-  ..'.', 

%*> 


fc^ 


and  the  actual  jiath  caw  he  at  ..nee  plotted. 

i  he   \,ihii-   of  y    ./,,//•  can     easily    he    found    !,;•  aphically. 

Tims.  p|,,t  the  radii  ('A'.  ('.I/,.  (U/,  .  .  .  ( '.1/ as  al)sci-.>a  .  and 
tlie  correspondinj,'  sfctional  areas  of  the  passaf^^e  at  A",  .!/  . 
.1/,  ...  .1/  as  ordinate^,  joining;  the  upjjer  ends  of  these 
ordinates  b\-  a  suitable  cnr\e.  the  area  between  tlii^  ciir\e.  Hie 
extreme  or»lii\ates  and  the  line  of  absiiss.e  is  evidently  the 
volume  re<|nir.'<l  Thi-  .irr.i  mav  be  determined  with  .i  pl.i- 
nimeter. 

3    Classification  of  Turbines.—  lln    character  ot  the  con- 
struction   ,,f   turbn\e^    lias    leil    to    their     beinj;     classified    as 
I  1  Kadial-llou    turbin.s;   ,  • ,    \  y  i,,l-n,,„   tiirhii-.-        •;  i    .\Ii\ctl- 
l1ow  turbines. 

The  water  ma\-  act  wlmliv  In  pressure  or  wholly  by  im- 
pulse, or  partly  bj-  jiressure  and  jiartly  by  iinpuUe,  or  by 
reaction.      |n  ])ressur<-  wheels  tlu  >>  ii.  i -p,issaj;es  are  not  com- 


THh  HU  RNHYRi^S    TIRMSF.. 


491 


plctciv  tilled  a-^  in  rcacti'm  wliccls.  hi  iini)iii-.i-  uliccN  the 
water  -picail-  out  in  all  dniclinn-.  wliilc  in  piessviro  ami 
rcaitinn  wlucN  tiic  water  flows  off  on  one  >i(lc  on!_\  . 

In  I\,i<ih)/-flf7.'  turbine--  the  water  flows  throuL;ii  the  wheel 
in  a  ilirection  at  ri^iht  ani^'le-  to  the  axis  of  rotation  and 
approxiniiitelv  radial.  I'he  two  special  types  of  tiii->  class  are 
the  Oitl;K'ai;i-th'W  tiirhine.  invente<i  In-  l"ourtie\ron.  and  the 
/ii:,<,ini-/!fU  <>r  I'l'iiiX  turbine.  in\ented  hv  ].\mv^  Tlionison. 
in  the  outward-flow  turbine,  l-ii;-.  jS;;  and  ::S0,  tiie  water 
enter-  a  evlimlrieal  ehanilur  .md  is  led  li\-  means  of  fixed 
;;uide-bhules  outwards  from  the  axi-.  It  is  distributed  over 
tile  inii  t-surfaee.  ijas-o-  throu|.;h  the  curved  passaijes  of  an 
annular  wheel  closeh'  surroiindiiv;  the  chamber,  and  is  linally 
discharged  at  the  outer  nirf.ice.  The  uheel  works  tiest  wiien 
it  is  placed  clear  above  the  tail-water.  .\  -erioiis  practical 
defect  i-  tile  difficult)'  of  coiistructiii'^'  a  suitable  sluice  f  )r  re;,'u- 
latin-  the  -uppl\-  over  the  inlet-surface.  When  the  water  is 
insufficient  to  work  the  turbine  at  its  hill  i)ouer.  the  exit 
oi)eninL;s  ma\  be  closed  to  .mv  ntpiired  extent  b\  lowering 
.-I  cylindrical  sluice. 

/  «  ell-desi','ned  turbine  of  thi-  t>  pe  ;.,'ives  an  efficiency  of 
-(^  ])er  cent,  and  tlit  m.ixinium  efficiency  is  .ihout  So  per  cent, 
but  the  efficieiicv  is  considerabl>-  dirninished  by  closin-;  the 
sluice.  l-"ourne\ron  was  led  to  the  desi^^n  of  this  turbine  by 
observinj^  the  excessive  loss  of  i-nerj;)-  in  the  ordinar\  Scotch 
turbine,  or  reaction  wheel,  and  introduced  i^uide-blades  in 
order  to  K'^^'  ^''^"  "i't"-'  •'"  initial  forward  velocity  and  thus 
cause  a  diminution  of  the  velocit.v  ■!  the  wattr  leaving;  the 
outlet-surface. 

l<o\-<ien  s  turbine  is  ,1  modifi  ation  of  the  1  ourncjron. 
Ihe  water  is  conducted  to  the  ^'uide-blades.  wluch  are  inclined 
so  as  t<«  receive  the  water  tanj^entialh  .  throii^;li  a  truncated 
cone;  and  the  watir  thus  .i.  ip'Mes  .1  ;.,n-.«du,(ll\-  increasin^J 
velocity  toj;ether  with  a  spiral  motion.  Ihe  wiieel.  a;^Min.  is 
surrounded  by  a  ./////,.v,/-  which  expands  outwardb'   i"d  ^^  hich 


>i 


THi:  HOYnh:\  tirhim:. 


^l.-ul.l    be    cmplctdy    suhnicr.L;.-.!.       T|,c    uat.r    thrn    flows 
tlir.mgh  the  wlR-cl  uith  an  incrcasi.l  veinuty  an,l 

l-u..    2h';. 


p.i^sfs  away 


ttte 


\^A 


Fi. 


:%. 


tl.mu;,!,  tluMliffusnr  with  a  velocity  wind,  ^^r.ulnally  din.inishes. 
I  here  is  sai.l  t.,  I,,.  ;,  jr;,;,,  ,,f  ,  p^.^  ^.^.,,j  effecte.l  l.\  tliis  a 


.irraHKf" 


THE    lOKIHS    ILKBIM:. 


4>)3 


nicnt,  while  Jviydcn   i.i.iiinc( 
cicticy  of  SS  per  cent. 


1  r..i-  h 


turl 


line   ,m   ttti- 


In  tile  /u:.i>r,i-f7i>:^-  oY   /'('r/i-.i  turbine.   I''i,4>.  2S7,   JSS,  and 


l-"i.;.  23; 


,  ^--^ 

—     — 1 

ij 

j 

— "                  1 

1 

til 


.lieel 


1^  en- 
closed ill  an  .mmihir 
.space,  into  wliich  the 
Uiiter  tlows  tliroii-h  mio 
or  more  pipes,  and  is 
usn.dl)-  distrihuteti  o\cr 
tile  iiiKt-siirface  of  the 
wheel  1>\-  nieiins  of  four 


'uide-l)ladt. 


The  water 


enters    tile   wheel.    t1ows 
tow  ard-.  the  space  around 


til 


e    ,i\is,    ami    is    tlierc 


ilischar;4ed.  I'his  tur- 
hiiu  possesses  the  ^'reat 
ad\  iiitai;e  tiiat  there  is 
ample  sp.ice  outside   the 


AO-i 


IHh    .-{.M.-H.-hlOH-    liKHlM-. 


,M 


Axuil.ji.nc  turhiiu's.  Vv^.  290.  arc  aUo  known  as  /\in,/lti~ 
and  / hK,;ruuin/-jl„:,-  t.mh\nc>  Mu\  arc  ■^MniLtiincs  callci!  l)y  tlm 
names  nf  the  invent.,!-.  J.,n\al  and  1-untain.  ,  In  these  the 
water  passes  .iowiiward  tlir.ni.L,di  an  annular  casiiit;  in  a  (hrec- 
tinn  i)arallcl  to  the  axis  of  rotation.  .i'..l  is  .iistributed  hy  means 
ot  .t,aiidc-blades  over  the  inlet-siirlacc  of  an  adjaeent  wlieel.  It 
enters  the  uliecl-passai^cs  and  i>  fin.dly  dischar-ed  \crtieally, 
or  ncarl\-  so.  .it  tlie  oiitlct-surfaee.  'ilie  sluicc-rciiulation.s  arc 
worse  even  than  in  the  case  of  an  outward-tlow  turbine,  Init 
tliere  is  this  advantai^e.  tiiat  tile  turbine  may  be  placed  either 
below  the  t.iil-uater.  or.  if  Mipphed  with  a  suction-pipe,  at  any 
point  not  exceeding;  30  tt.  above  the  l.iil-uatcr. 


tf 


\ 


Ji 


\zi 


X 


FlC.    2(|0. 

If  a  turbine  is  desione.!  so  tliat  the  pressure  .it  the  clearance 
between  the  casin-  and  the  wheel  is  ml.  ,ind  uith  eurved 
passa-cs  in  tlie  form  of  a  heel\-  devi.itcd  stream,  it  becomes 
Mliat  is  called  .1  Lnmf  turbine.  In  its  norma!  condition  of 
uorkin-  it  i.  .m  Impulse  turbine,  but  «hcn  drowned  it  is  a 
Reaction  turbine,  with  a  small  pressure  at  the  clearance.  i'or 
moderate  falls  with  a  \aryin-  supply  its  avcratjc  efficieiuy  is 
lii^dier  than  that  of  a  pressure  turbine. 

I  he  .y|^,,^-  or  ('"w/vV/o/-/,;:.- (Schick- >  turbine  is  a  com 
bination  ol  the  radial  .in<l   .1x1, il  t>pe..       The  water  enters  in  a 


f^ 


I  hi;     -IXi.-U   I  iim      ILRHiNH. 


4<-5 


1' 


ncarlv  ra.li;il  direct,. -n  aiul  kav.-  n,  a  dnccti-.n  apprnxuiiatcly 
iralicl   tn   tlu-  axi-  of  r..tat.on         Ih..   tyi.c  ..l   turhnu-  admit-; 
.,ta  -.H)<!  111. hIc  of  rc-ulation  and  i-.  clica])  to  o.nstnict. 

riK'  Suam  turbine  i<  a  .  o.nl.inatiun  of  the  inward-  and 
.,x,al-ri  -w  IV1.L-.  Tlu;  vane  inlet-lips  are  verticd  opposite  tile 
',,uide-l)!adJs,  an.l  al  the  outle^t  the  vanes  are  bent  into  ., 
quadrant  of  a  cude.  An  etlkiency  of  SS  per  cent  has  been 
claimed  for  this  turbine  under  a  full  load. 

Comfarison  .f  (h>tu;u;i-;?,n.'  Thr/'iiNS. -V.mrn.-ynm  deals 
with  a  varying    supply  of  water   by  means  nf  a  eireular  -huce. 
vvhieh    can    be    .nade    to    c!<.;e    off  any    required    portion    <.t 
the    wheel.      .\    similar    arrangement    may    be  added    to    the 
Cadiat   turbine,   which    i^   of  the   ..utward-tlow  type  and   .-   fed 
fr„m    above    thr..u:^d,    a   cylindrical    reservoir,    the    ni'I-er   and 
lower  ed-e-  of  the  reservoir  bein-  rounded  to  diminish  the  los- 
,hie  to  contraction.      The  objection   to   sluices  ..f  this  kin.l   is 
that  the  passi-es  no  lon-cr  rim  full  uhen  the  inlet  ..rihces  arc 
p.rtially  closed  and  there  is  therelore  a  considerable  .liminution 
,.f  cfTicioncv.      In  the  Whitelaw  turbine,  p.  37?.  tiiis  .lifficulty 
can  bo  obviated  bx'  chan-in-  the  outlet  instead  of  the  inlet  area. 
Tlie  abs-nce  ..f-ui.les  in  the  Cadiat  and  Whitelaw  turbines 
,„  d<e  tluir  construction  somewhat  simpler,  but  their  efficiency 
is  comparativelv  small,  that   of  tlir  Cadiat  bein-  about  u;  l-er 
cult    while  the  emcienc.v  of  the  Whitel.iw  turbine   vanes  from 
;o  to  60  ].er  cent.     <  )n  the  other  hand,  the  l-ourne>  ron  turbine 
h.is  an  efficiencx-  of  more  than  70  p.r  cent  and  is  mechanically 
a  much   more    i-erfect  machine.       The   -uides    in   the    turbine 
render    it    possible  t..  ntili/e   almost  the  whole  of  the  energy  of 
the    water    either    bv    e<iuali/.in-    the    peripheral    and    relativ, 
speeds  at  the  outlet,  or  bv  m.ikin-  the  absolute  velocity  at  the 
nutlet    radial.        The    l-ourneyron     and     Ca.'iat     turbines    are 
speciallv  ad.ipted   for.    la,-e  sui-ply  of  water  an.l   a   moderate 
fall    sav  not  exceedin-  about    ^,0  ft.,  while   the  Whitelaw  tur- 
l.nu.s  are  found    more  useful  for  a   small  supply  ot   water  and  a 
hi-h  fall. 


A'i6 


THEORY  Ol-    ILKlilShS. 


■m 


Fic;.   291.  — KnlarKed  Portion  of  St-itioti  through  .V»',  V\^.  2^-;. 


^      3     ~ 


Fig.   292.  — Enlargr.l  I'ortion  of  Section  ihrmi^h  .V  )•,  Kii;.  2^5, 


M 


Fic.    293.  — Enlarged  I'ortion  of  .1  (.vlindrical  Sectl.in  .\y.  Fig.  290, 
Developed  in  I'laiir  of  I'.qicr. 


•fp- 


THEORY    Oh    ILKlilSES. 


40  7 


4.  Theory  of  Turbines  lii;-.  2')\.  2<j2.  ami  29.V  .  — 1  )iiintc 
iinsar(l-i1n>\  ,  .  mtu  .inl-llM\\ .  .md  axuil-flow  .  turbines  b\-  I.  I'.. 
(  ).    1"..  ami  A.    I'.,  r<.'si)e(.tivcl\-. 

Let  /•   .  r.  be  the  radii  of  the  wheel  iiil.'t-  ami  <uitlet-.iirfnccs 

..fan  1.   I-.  or  <  >.    1. 
Let  /•  .  r,  be   the  iiuter  ami  inner  radii   <>t"  the  wheel   inlet- 
siirl'ace  of  an  A.   !". 


Let    A'    be    the    nu:an     t 


adu>.  (= ''";'')  "<■  ■'"  •■^-  '■'• 


assumed  constant  throughout. 
Let    .-J,,    -1,   lie   the   areas   of  the   wheel   iidet-  and    outlet- 

oriliees. 
Let  ,/,  .  </.,  be  the  depths  of  the  -a:ne  in  an  L  I",  or  < ).  1". 
Let  </, ,  </,  be  the  w  idtlis  >>!  the  same  in  an  A.   I- . 
Let  //  be  t!ic  dejith  of  the  wheel  in  an  A.  F. 
Lei  //,  be   the  effective    head  over  the    inlet-surface   of  the 
wheel.       This  i-  the  total  head   over  the  inlet- 
surface   diminished   bv  the  liead  consumed   in 
frictional  resistance  in  the  supplN-channel.  and 
In    {])'■  he, id  lo^t    in  bends,  sudden  Lh.in,i;es  of 
section,  etc. 
i'hen   //,  -4-  //  i-  the  total  lie.ul  over  the  outlet  of  .111  A.   I". 
ax.iilable  for  work. 

Let  // .  be  tlu'  fdl  irom  the  outlet-surface   to  the   surface  of 
the  water   in   the  tail-race.      If  the   turbine   is 
subnier;;ed.  then  //_,  is  negative. 
Lot   T'  .  7',  be   the   absolute   \elocities   of  the    watei    at   the 

inlet-  .imi  outlet-surfaces. 
Let  II  .  II.  be  the  absolute  velocities  f)f  the  inlet-  and  outlet- 
surfaces.      In  an  .\.    L    tarbinc  !.\  =  ".■ 
1  Lt  T  ,    /',  bi    the  \<  JocitK'-  of  the  water   relativel_\'  to  the 
wheel  at  the  inUt-  and  ontlet-siirfaccs. 

It,  14, 


Let  th.e  angular  \elocity  ot  the  wheel         i.t 


2i 

1         fi 

.et  /;  dcsi'Mi.ite  the  hvdraulic  <  fficiency  of  the  turbine. 


49.S 


THI.ORY   or   TLRHISES. 


\> 


l.ct  the  ^\atcr  (.-ntcr  llir  wluol  in  the  direction  ii,  ,  uKikinj^- 
.111  an-lr  )'  with  the  taiiL^eiit  ,/-/.  l'. ike  rft' to  represent  :\.  and 
(/</  to  repie-^ent  .v,.  t'Diliplete  tile  ])arallel(),L,M-ani  />(/.  Tlie 
side  (.'/'  ri'preseiits  l\  ,  and  in  order  that  tliere  iii.iy  be  //i>  s//ot/c 
,if  iiiti-ivui\  if/i  nuHt  he  tan;_;eMtial  to  tile  \  ane  at  <?.  Ai^ain. 
at  /  dr.iw  /V.  a  taiiyeiit  to  the  \,me.  .iml  /,r.  a  tani^'ent  to  thi- 
\\heer>  periphery- 
lake  /,;•■  and  //■  to  re])re-'ent  / ',  and  /^  respeetively.  Coni- 
])lotc  the  parallehiL;rani  .^'X'.  llio  dia^'onal  y//  must  represent 
in  direction  and  n  nitudo  the  alisolutc  velocity  r  ,  witli  which 
tile  water  leaves  the  \  heel.       Let  the  an^Me  ////-•  =  ')'. 

Dr.iu  , ///  peipendiciilar  to  if/,  and    ////  perpi  ndicular  to  _(,-■/•. 
1  ho  tiVi,i:;iiitiiil  component,  \i/..,  am  or  ///,  of  the  velocity 
o|  the  water  as  it  enters  or  leaves  the  wheel  is   termed  vehhitv 
i'/z,'/iir[  {:■.,.•). 

I  lie   rndial  component,  \i/.,  iw  or  ////,  of  the   \elocity  of 
the  water  as  it  enters  or  le.ives  the  \cheel    is  termed  vilocity  of 


Take 


am. 


=  lin. 


,  I 


It: 


•^v 


N 


I.et  the  an^le  />/{</  —   i  So '  —  a. 
•  et  the  anyle  .i,-;//-  ~  180    —  /i. 

Tluis  <t  and  /J  are  the  anodes  wliich  the  vane  (or  hkidei  tij) 
(<M-  lips)  make  with  the  wheel's  jjeriplieries. 
Then,  at  the-  inlet-.-^iirrace, 


t' „     —  T    COS   I 


rff  COS 


^  )/  =:  iiiH  =  ,/,/  ±  dm  —  u^—\  '1  cos  ,t , 


■illl    '- 


r  w  --  / ',  sin 


(2) 


and  at  the  oiitU-t-viirlace. 

V,,.    —  -,\  cos  6  —  fu  —  fk  ±  kn  =  «,  —  ]\  cos  fi 


I','  ~  >'j  sin  6  =  //;/  —    /",  sin  /i. 


0) 
•      ■      (4) 


l-vJ 


THIOl^Y  OF   TiRmNFS. 


499 


Let  Q  be  tlie  quantity  >A  watir  whiiii  pasm-,  pir  sccotul 
throii;j;h  the  turbine,  'i  iien.  ili>rei,rar(linL;  the  tliickn----^  (if  the 
vanes,  ///  <///  /.  /•".  cr  0.  /■'.  tiirhi)u' 

v/A,  =  V, 

anil  therefore 


2  7r,d,       Q    :  v/  .  2Tr,d,       v.  A, .    .     (5) 


■    ,       Q 


«r^ 


Also,     if,/,    =:z   d.,    =    ,/. 

1)1  (71!  .  1 .   /■'.  tiirhiuL 
V,.  A        v^  .  2tR  .  d,       Q       V,. 

anil  therefore 


V  r,. 


2tR  .  d        v^  A  ,     (f'j 


If,/,  =  ./,----</. 


Q 

Allowance  may  be  made  for  \aiie  thickness  as  follows: 
Let  "  be  the  an<4le   between   the  \ane  of  thickness  /.'('  am! 
the  wlieel's  pi-riphery  J/)'.     Then  tlie  space  occu- 
])ieil    bylhe\,ine    aloii;;    IJU'    uha-el's    pe|-iplier\-  is  ^, 

A 11  -  JlC  cosec  H. 

Let  It  be  the  number  of  the  f^uide-vanes,  ami  / 

their  thickness.  V\>..  2<u- 

Let  '/|  he-    the    inimhe;-    of  tile   wheel-\anes,   and    Z,  ,   /..their 
thickni'ss  at  the  inlet-    and  outht-surf.ices   respec- 
tively 
'I'lien.  in  a  r.ulial-llow  turbine. 


y1,  =       d^\27ti\  —  III  cosec  ;'  —  ;//,  cosec  tt\        .      (7) 


>i 


500 


THFOKY   OF   Tl'RRINFS. 


10 


(i.,]2-jr,  —   w  /.,  msi'C  fi\. 


(8) 


•,',,  heinj^  a  fraction  dcpetidini,'  on  practical  considerations. 

In  an  a.\ial-tlo\\-  turbine  A'  is  to  be  substituted  tor  r,  anti  r.^ 

Ill  the  \,iiues  ot"  .  /  ^  and  .1  ,. 

/,,  may  he  made  equal  to  // 4-  1   nr  '/ -f-  2. 

Work    ami    I-.ffwiciu-v. — As    the    water    Hows    throuj^^h    the 

wheel,  let  v  be   the  velocity  of  flow  at  anj-    point  .V  distant  >■ 


a— ^— 


Fic.  29J. 


('=  (>.\'~>  from  tlie  axis  O.  an/i  let  /  be   the  lenLjtli  of  the  per- 
pendicular from  f  '  u])on  the  direction  ot  ,''.      'I'hen 


wQ 


—  momentum  of  moviiiL,''  mass  of  water 


15*. 


•^i 


M 


=  impulse  on   wheel 

=    /■'.     SU])pl)St'. 

Therefore,  also, 

— ^r'/>  =/•'/>=  moment   oi   coujile    producing''   rotation, 
and  the  useful  work  of  the  couple  per  second 


liwr 


THEORY  OF   riRHINF.S. 


SOI 


But   if  T.,,.  is  the  CMiiipdncnt  <>f  :•  at  A'  perpendicular  t"  tlie 

radial  lir.c  ('A', 

vp 
''■w  =  '•  COS  '^  --    ^.  . 

and  thcrcture  the  useful  wcirk  of  the  omple  pi-r  -eccnd 

—      -  v^roi. 
S 


Thus  i'l  an  1.   /■'.  ("•  ( '.   /■'.  turhinc 


the  useful  effect  at  inlet 


.-.,.  /■  O), 


wQ 


■'•«i. 


the  useful  eftect  at  outlet  —       ^  ;•„,  r,a>,   =— — Z'^  ?/., , 

and  the  i  >1  1  '  1     WiMiK    per   second  d(jne  b\-  the  water   on    tht. 
wlieel  Ijetween  uilet  and  outlet 


# 


wQ 

g  ' 
wQ 

g 


V,,  r,       v^  r,i(.>, 
v„.  u,       v„,  u,). 


The  i;i' I'll  IKM  \-  is  L^Mven  \i\  the  relation 

V  X  "''■'Q^^i  —  ^''^'  I'seUil  \\>irk  pe'-  s'-C. 


or 


;/gH,       V,,  u,  -  v„,  u. ,  .  .      .     .     (I  I J 

which    is   the   fundamental    e.pi.ition   governin;^    the    tlesign   of 
I.   V .  or  <).   V    turbines. 
In  ,1)1  A.  J'-  tni-bint 

ivQ     ,,,        i«'<3    , 

he  usehd  effect  at  niK-t      —  —^v.,.Ka>  ^   —V^u, 


th 


ff 


the  useful  effect  at  outlet  rr_      ^  :■,„"/<<.>  -  •  „.  «,  - 


5°^ 


THIOKY   ()/■    /I  kUlNFS. 


and    thr  I  mi  i  i,  \\<,]<y    jur  srLdi.l    .lone   by  llu;  u.itcr    on    tlic 


h 


5;| 


•;i 


wQ 

wQ 

g 


'V,„        V,,     -R,.' 


V„.     lU, 


(  12) 

1'3) 


or 


The  rItKicncy  is  ^'i\iii  hy  tin-  rcLition 

V  X  -u'QUI^  -f  ^1)  —  tlic  US'  lul  U'.ik  |itr  sec. 


■'g(H,   i  hi       .v,,;       v„.   )u, 


i.p 


\\lii^h    is   tile    iuml.micnt.il    t()ii,aiiin    ;.;m\  i  rni;-.;^    the   (lcsi"ti    df 
.\     1  .  liiiljini's, 

A^'ani.    i/isr,  '^,nf/i>/x'  /ni/riiu/ic   iisistinin  s,  c.u  li    ]ii.unii   <if 

w.iti-i   nn  li  a\  in  ;  the  tnrhinr  (.  ,11  ru's  ,iu  ,i\'     "'    lt.-ll)s.    of  i'n<T"\-. 

ll.ncc 

the  isi  1  I  I    WdKK  in  an  I     V    or  t  »     !■'    tmliinc 


wQ^H, 


tile  I  iirrcspdtKliiiL,'  1  II  |i   II  \i  \    inin;,;    I 


v- 

2gH, 


ami  the  I  s|  M  I    \\,  ,KK  HI  an  .A.    1      t 


<'5) 


ni  iiini' 


L. 

mm 

H^ 

^^^B..-^ '. 

f  \,\ 

WnS"*-' 

'    I'-' 

^BmiSKii . 

J!     ll-.l 

/I 


;i    K,|i» 


wQi^K,   ;  h 


2g 


the  inrrfsiH.iKlinn  liiK  ii:\(  v  lidnj;  i 


2g  H^    :    h, 


•/-) 


lHht'h')'   < '/     / '-  KlilNFS.  5 <^  .1 

A-^^viniiii.L,'  thai  tiiL-  velocity  .ifuliiil  ai  outk-t.  viz.,  T\„  ,  i- 
nil  and  tiiat  //  i-  the  portion  nf  // ,  ,  .t  of  // .  •  //.  uluch  i^ 
traii'ifornK'd  into  usctnl  \sork.  lluii 

whieli  may  l)c  wri'.t-.n  in  the  form 


«,        /        ti,  "'r  .        \ 

— 2^  I — ,J cot  «|, 


a  (luadratic  s^MVin^' 


■I       _ 


cot  rr 


\  ..^//       V^// 


This   ri'-ull   ii.is   biin  cmiiloycci  m    iMi-parini;  the  follow  ni^' 
Tabic  ot  val. us   ot       "j       cornsix.n.lin^'  to   (liffcrcnl   vahK-   of 


4,C// 


anil  o|  ,) 


"«// 


UJ*   i  :>••  1  ••»• 


t^o! 

1 

«.l89 

i 
I.6IH 

1  3»<) 

I.I43 

0.9 

i.(it<) 

3047 

«.547 

1.188 

1.128 

0.8 1 

J,  289 

l-'>"9 

1-477 

I.WX' 

I.II4 

0.7 ' 

sqsa 

l.77t' 

l.4<x> 

1.325 

l.CK)5 

0.6  I 

3.631 

1.647 

1.344 

I  188 

1 .083 

1 

o.S 

2.301 

1  »».1 

I.»Sl 

1.155 

I.OfM) 

I 

0.4S 

2.r45 

l.4'>3 

1.250 

I.I3'> 

I.<j63 

1 

0.4 

•  W 

l.4<)< 

l.23<> 

1. 133 

I  055 

CIS 

1.S47 

l.34'> 

I.I9» 

1  U17 

1.048 

"l 

I.?''? 

1.393 

l.lM 

I.IKJ" 

I.U4I 

i..a«i 

H»K) 

1.^40 

t.iia 

1.074 

1  034 

i      ' 

o.a 

I  44" 

I.I«8 

1.105 

I.')!*) 

1.027  '    1 

IMS 

l.uH 

I  n» 

i  1.07* 

1  044 

i.oao '    1 

0.  Ill 

1.204 

1    IK|II 

!  i.iu' 

1  1)29 

t.OIJ 

I 

.874 ' 

.887 

.l>0<) 

.90* 

qj3 

•  '135 

.<)43 

•  049 


■  75» 

•773 
•  795 
.819 

.843 
.886 

.879 
.8<)U 


.954  I    .90a 


.967 
')?1 


.<i3o 
943 
.>)66 

.97a 


.618 

.647 
.'•77 
.7"9 

•  744 
.781 
.8i» 

.8au 
.84U 

.8<KJ 
.881 

•  90! 
.9»7 
.9*1 


457'  ■«!« 

.488  .371 

.534  .304 

">3  -339 

.(«/  .3S3 

657  -435 

.683  .4<5 

.7«»  -499 

•744  J4« 

.773  -SX' 

8o<>   .656 

I  .842'  .'•Q4 

I  •8785  .75* 

I  .917I  -^JO 


S04 


f\.-tMri.FS. 


It 


A;i,;uaiicc  may  br  tnadc  U,v  the  pri-.^-pal    hydraulic  rosist- 
aiuc<  h\-  takiiiLT 


Vl 


X 


N 


/.-tV  •■'■*  '•'•-■  ''"'■^'^  "t"  lica.l  hrluri-  cntcriliL,'  the  uhecl  ami 

/i,j.   •'■^  '''^'  ''"-  "•  '"•'"I  l>i-)"rc  rntcrm-   in  the  whf(l-i)a-.sa"cs 
'!  lull   ihe  t.ital   j.r,-,  .,1  lica.l 

'■  •'        /  v     -■  • 
=  /iT|^,-fy,j^+ 3'^, n9) 

'I'lic  \alucs  ofth.'  empirual  i-.Kfficiciit^  /',  an.l  /,  ina\-  van-, 
the  l.iiiuer  tiMiii    tij;  t..    ju,  an- 1  the  latter  Iimiii  .10  to,  jo. 

i;.\.  I.  W.itir  eiifrs  an  ().  l\  uirbiiie  ol   i',  n.  exterior  ami    if  ft.  in- 
terior cliaiMi-t.T  with  a  whirling  velocity  of  20  ft.  [wr  seco.ui.  and   leaves 
in  the  reverse  direction   witli  a  whirling  velocity  of   10  ft.  per  second 
The  wheel  makes  .'40  revoluti-ins  per  minute.     Kind  the  useful  liead 


"    1%  .  240 

(X) 


2-  ft.  |)er  sec ., 
"1        -/',        44  ft    per  se< 
Tlieii.  if  //  IS  the  useful  head, 

U'{)//  =  work  (lone  in  driving;  the  whetl 


ant] 


A' 

H  =  i7i  It. 


(       lop  )     -  u'Q 


8«o 
J-' 


Kx.  2.  A  turbine  with  a  radial  inlei-lip  receives  10  cii.  ft  of  water 
per  second  at  a  radius  of  j  ft.  .,nd  makes  105  .evolutions  ,kt  minute 
The  water  enters  at  W  with  the  wheels  periphery,  and  leaves  without 
velocity  of  wliirl.  rf  the  eiricienev  is  «M.  find  the  elfective  head  and  the 
II. I*.  o(  the  turbine. 
Since  It  =  >/j  . 


f»   ^  «. 


*    4.  IPS 


EX/IMPLES. 


505 


and 


Til.' 


.88  -  tlie  ofTicii-iuy 
//i  =  17.1X75  ft- 


Ux- 


484 


^//,   ~32//,      32/// 


111'.  =  ?-l-J°  X  .88//,  =  17.1X75. 


\'.\  ',.  The  wheel  of  iin  .\.  F.  lurbme  of  ',  ft.  interior  riiameter  has  a 
6-in.  uiihh  I  f  orilice  opciiiiiy  a\\i\  l^  1  It.  ilrcji.  It  f)asses  33  cii,  ft,  '.f 
w.iter  [nT  second  under  tlie  head  of  24  ft.  ovei  the  iidet.  and  the  water 
leaves  the  wheel  in  a  direction  ^'i\en  by  cosec  fi  =  1.015.  Determine  the 
etriciencv. 

liy  the  condition  of  conlinniiv, 

'[(4'  -  3'i''."  =  33  =  5JV. 


or 
Therefore 


iv"  =  ''  ft.  per  see. 

Tt  ^  7'."  (  osec  ■>       o  A    1.015  -    6.09  ft.  per  sec 
r.'  (6.09)' 


the  ellic  n-iK  v  —   1  — 


^->H  +  n 


l(>oo 


=  .V7'')8. 


rii 


III' 


62i  33 

3-  550 


X  :,  X  .9768  =  2.929. 


K.\.  4.  I'md  the  i>utlet  hp-ati^le  i/i)  from  the  l.ji'wiii^;  data:  railius 
to  inlel  =  /tt'/ic  that  to  outlet  .surface;  linear  speed  of  miet  surface  = 
iiiit-hiUf  that  ei|uiv.ilent  to  the  effective  head;  inlet  vtloctiv  of  Jl^no  — 
cne-tiiihtli  of  that  equivalent  to  the  effective  head  ;  sectional  area  of 
waterway  is  Constant  fruiii  inlet  to  outlet  :  the  water  leaves  without  ve- 
locity of  whirl. 


Th 


«  \<lh       .(  Ml. 


and 


I'y  c.mditi  in  ot  coiuininty, 

I 


Hence 


cot  (i 


-•♦  //, 


♦  //. 


=  J. 


qoO 


HXAMrLES. 


,1 


■*ris 


Js:< 


Kx.  5.  The  wheel  nl  .1  turbi..e,  ,  assin^  10  iii.  (t.  nf  water  per  secuiui 
under  .1  head  of  32  ft.,  is  6  ins.  deep  and  Us  inK't-surlace  has  a  diameter 
I  if  .;  feet.  The  iidet-lip  is  radial  and  tlie  ellK  leney  may  he  assumed  to 
he  unilv.     Find  the  ^iiide-vanc  lip-anj;le  and  the  power  of  the  lurbine. 


I  =  the  ellu'iencv  = 


«iJ'« 


3-  X  3: 

Therefore  «,  =  3;  ft.  per  sec.  =  J 

Hy  coiKiition  of  (ontiniiity, 
7.2.1.  ;./  =  10, 
3 


1024' 


Hcni  e 


TV'  =  ~  =  3  m  ft    p'-rser. 

tan  ^  =  '  '  =  —  =   09943. 
II         32  •'^■'J' 


and 


y  =  5°  41'. 


riie  II.  p. 


32 

550 


Kx.fi.  Ill  an  inipnlse  r.niial-flow  tiirhim- the  inlet- and  oullel-orifire 
are.is  are  equal  and  the  waliT  leaves  without  veloriiy  of  whirl.  Disre- 
','ardiiit;  hydraulic  resistances,  show  that  the  velocity  of  w  hirl  is  cos'  y. 

Hv  conoition  of  continiiitv, 


,/,?/,  =  Atv"  —  Q  =  AiTr  =  ^/ifi  sin  y. 


Therefore 


7',  =  -',  sin  y. 


and 


the  elhi  lencv  =   I 


V,' 


I    —  sin'  y  —  cos'  y. 


Ex.  7.  In  a  radial-flow  impulse  turbine  the  peripheral  and  rel.itive 
speeds  at  outlet  an-  ecpial.  Show  th.ii  the  direction  of  the  water  at 
inlet  hisecis  the  an^le  between  the  rim  and  the  inlet-lip.  Also  show 
that  riVi  sin  zy  =  r,'',/,s\n  ti. 

Hut  I'l  =  u,  ,     .iiid  therefore     J'l  =  u,, 

so  thai  zy  -    I  So        ,t. 

Hv  condition  of  rontinirtv, 

inv,tlx7,'  =  <J  -  iiTf,,/,  -  -;", 
nr  r,i/,  .  /'i  sin  <r  :^  >,</,  .  /',  sm  ff, 


or 
or 


r,(/,M|  Sin  2y       r-itf,  .  lit  sin  [S  -  r,,f,  _  -«,  sin  fi. 

'  ''1 
r,*i/,  sin  21'       /  ,'</,  sin  fS, 


THi.OKY    Oh    II  KlilN'iS. 


507 


Appluation   of   7'.'rri,c//i '  s    Prntapl,  .  ^\^        .     •     arc    lli<- 

prcssurc-licaiU    ,il    the    iiilcl-     and    1  iut!i.t---urr.n.  i"-    ■  i|    a    Uirl>iiU' 

P  —  P , 

wheel,  the  LiTceti\e  ht-ad  n\er  the  inlet-oritue^  !■-   //,  —      '      - -". 

Hence,  dtsri:^iiidiii:^  lt_Vtir,v(lii  risisliVUt  s , 

vr      _        p,       p, 


IN     A      Kl  A'    I  |M\       rCKIlIM 


^g 


H, 


w 


30j 


III  tuihiiu"^  'if  tln'  nnpidsi  type  p^  --/.■  ■""!  'l>e  watei'  i-< 
u^uaiK  umler  alniii-.pherie  pressure  niily  l)iitli  at  miet  aiiil 
outlet.       riius 


K2\  \ 


V 

i\     \N    iMi'i  \>\     n  Kr.iM       '    --  H 

2g 

Allnwanee    may  he    iM.ule  lur  tile    in,-,    .  )f   head  at    eiltraiiee 

mill  the  wheel  In' sub^litutmij      .,    ~  for  -'     in  tliese  two  eciiia- 

tions,  tile    a\er;iijc  value    of  the    enipirit.d    .  niftu  ietit    .      heuii^ 
ahdiit  .<j4'^,  or  cr  --    'j. 

Application  i>f  liiinouilli  s  Pi  111,1  pU  . 

\\      \     1.1    \i    I  !•  i\     I       !•'      I  IK    (  ).     !■".      I  I   KI:1\K 

Ti'     '      2g         U>    '      2g  2X'' 

the  last  term  liem;,;  the  \\  ork   |ier  ])oiiiid  1  if  \<  ati  r  due  to  c  entrif- 
u'jiil  force.       Tlierefore 


y^-r-     «,._«,'/>,_/, 


-A' 


2^ 


•       •       (22) 


1 


or 


2g  "^       2g  ag 


H, 


(23) 


•^ 


■    i 


5°^  THFORY   OF    TIRRINES. 

uliicli  m.i\-  akn  he  urittcii  in  llic  form 


U|V,  COS  r        V,-  - 

g  2g 


u,- 


H,. 


since,  from  the  tri;ini;li'  ,ti</. 

In     AV     IMl'l    I  -.1       I.     ]•■.     (  )K     (  ).     ]•■       I  I   Kiiivi.; 

/>,  -  /_, ,      .111.1      e',J  =  2i;;H^. 
'riicrcrnre  ci].     2},\  hcconies 

f   '  ■ »  T  •  ■>  1  1 


■   r24) 


(25» 


2.< 


^A- 


(26, 


In'    an    1      1-".    MKiiiM-    II.    .  II,   and   tin-  term   ^ " 

2jr 


-       IS 


Hii^iUr.t-.  IIi'tKeei].  J  :  shows  that  a^  the  illK't  Veh  icit\-  r' 
increases  nr  iimiHiisiu>  the  -jxed  <<\  the  turbine  diininishes  or 
nicreases.  and  that  theretore  the  H'ntrihi;;.il  lorce  tends  to 
mamtaiii  a  stead\- motion.  .\  ihminiition  in  .  also  necessarily 
leads  to  a  correspondini;  diminution  m  th.e  loss  of  head  <lue  to 
hydraulic  resist, mces.       I'oi  these  re,is,,ns   the  lentrlhl•^d  h.-.id 


■^'"'iild  he  m.ule  .is  l.iri^i    .is  is  ]ii  acticahle,  ,ind  the  r.itio  '  '    — 
IS  iisu-illy  m.ide  .  ijual   to   j. 


I    _     1 


!■■-    \^   ' '.    1'     II  ui;iM.  //,  •■  II.  ,ind   the   term  'H 


K'  -  w. 


IS 


fositir,.  llellCi  the  .|)eed  of  the  turhllle  iiu  re.lses  .uid 
diminishes  with  ,  ,-.  .md  th.  i  entrifuL;,Ll  torce  is  adverse  to  stead\ 
motion,  tending'  both  to  .ui-menl  .i  v.iriation  fiom  the  normal 
speed  ,ind  to  incre.ise  friction. d  losses  of  head.  Tlic  ceiitrif- 
ii^al  head  should  tlieietoie  be  ni.ide  as  small  as  is  practicable, 

and  .1  common  \.iiue  ot  the  ratio    -  =        is  - 

"i        >■,       < 
In    a    ki  a< HON    A.    V.    Tfi<nl\K    eat  h    ihiid    particle    in 
l>assin^;  from  inlet  to  outlet  remains  ,it  the  same  liistance  from 


THHORY   Of    TlRlilMS. 


309 


the  axis.  an<l  tluTct' nr  n..\\"rk  i-  il"n.-  by  >.  <  ntrifu-al  furcc, 
hut  an  adi'iitii'iial  luail,  //,  ciiual  t"  tin-  .Icptli  ■ 'i  the  whcrl,  is 
L;anicil.       Then 


'V-  /r^/,  -  A 

rhcrcliinj 


1  I       ■  'J  tr 


i-/"i 


V,-'   ,    V/       V, 

2g  "  2g 

which  ina\-  aN"  !n-  written  in  the  |..rni 
U,V,  COS  }■        V.    —   u. 
g  ■^  2g 


H        h, 


(2.S) 


(30) 


+     -  _    -        H,  +  h,  ,      .      .     (29) 

I\     A\     IMIl   I  --1      A.     I'.      I  IkUlM'. 

/,        ,", .      and      .-,■'  -  2x-/f^. 
■rilcielure  eq.    iJSi  heeonies 

V    -  V  ' 

^-  '         h.       .     . 

2g 

In  order  t"  -cMire  the  a<UaiitaL;e>  <>\  n  ntrifui^.il  t'oree, 
Helan-ei  i.r'i]i"^e,l  that  th<-  w  heel -pas^a^'cs  shi)uUi  he  so  tornu  ,1 
that    th.     ]iith  .i|    a  tlni.l    particle  u  ,.iild  }j;radiially  ai)j)ro;icli  the 

a\i^  I  il  VI  itation. 

/..■/>  u<r  'Jif\  .\il,i^lis  — 'I'lie  ailt;les  ,.  and  (i  wllicll  the 
wheil  hlade  tip^  at  inlet  anil  outlet  in.iUe  with  lli<-  win  el's 
|)eill)llerie>    are    L;LMKiall\    ohtaillCil  as  follow  s . 

l-"roni  the  triangle  ani, 

sin  (a  +  y)^  «,  ^  ^,^,^  ^^   ^  ^.,,^  ^  ^.^  ^^ 
sill  <r  ."•, 

and   iheiefore 

cni  dSo"  -  a)  ~  ~  cot  It  -  cot  y  —  -'  cosec  y.     (3 1) 


5'0  THHORY   Oh    1 1  RIUM:S. 

l-'rciu  tiu-  ti-ianL;lc  //•//, 
>ii\  1/^  -r  <^         u.. 


sill  li 


•■(IS  'J  -|-  I.  ( it  (i  sill  (?, 


•  iiui  th<-!i|(ir(. 


cot   (iSo     —    //)  (.,  ,t   /y         cnt   a'     -  — '  coscc   ff.        (32) 


VI 


(.'//,////, I// ,v    (,'i'riri/in:^'-    ///,■    l-.tYumuy    cf'    Tiirhiius.  — Tlu- 
wilok'   iif  tllf  \\,ltrr's    cllcri;\-  sllnuld.    if   im^sihU-,    W-    clll[)loyc(i 
in  liniiiL;-  u>rtul  work  >^\\  the  ulucl,  .1111!  tlic  w  alcr  -.li.piild  tlicix-- 
i'ln-    lca\i-    till'    wlux'l    witlimit    \cl(n'ity.    or   ;■,   slmnlil    In-    nil, 
1  Ir.s    i;i>iuiitiMii    v.uiii'it    III  iniir^c  hi     rc.ili/cil  111  pi-avtiic,   as  iii> 
watrr   udulil   tiu-ti    ]ia->   tIirn!:L;li    tin-    \Unt.l   .iml    i. .  in-'(]U(.iitI\ 
nu  work  tould  he  liniu'.       I'm'  jiiii])i)scs  of  ttTKiciic)-  it  i-  iisLial 
til  make  .-^  >mall  by  aduptiiiL;  hir-  nf  tlu-  f. ilhiu  iiv^^r  liNpothcscs: 
1  I  I  m  k  tli(U  llii-  i\locity  of  whirl  at  oiitl,t  is  nil, 
I  IK  tluit  at   t/ii-  ,:i(tlit  tlu    relative  vflncitv  of  the  water 
aiiii  the  peripheral  linear    -eeloeitv  of  the   leheel  are 
e</iial. 
I'IK'-I   consider  tlic  Ii\|H)tIu'sis    "iliat  the  \c!<.(.it\   .  if  whirl 
at  nutlet  is  nil. ' '       Tii,  n 


(33) 


/.  /■■, 


O.F. 


A.F. 


Fi'!.  2<>6.  Fir..  297, 

'Iluis    tile    direction   of  .-^  is   radial   in  an    I      I      or  O.    F. 
Itirhnie,   |-"i<,->,   2(/.  and  2y;.  and  vertical   in   an    A     T    fnlune 


vkS^m^ 


THHOR)'   Ol     II  KHISrS. 


Vk',.   2<)S.  and   therefore   the  ,uil;K'  ////  (        '>  >  in  tlie   (luth't  tri- 
aiif^le  of  \eh.)cities  must  be  a  rii,'ht  aiii^le,       lieiKe 


mil 


!■._,  =  rv     =  ".  t.ui  /i    -    r.^  MU  /i,     .      .      .      {341 

V,/  —  u.-  —  I'r  -'-'-  II.,'  tan-  li (35) 

Also,  cq.   {l)  g.vcs 

f,  sin  ;'. /,  =  r'^./_,  —  11.^  tan  /i. /_,.    .      .      .      {yj) 

(iilhial   l\dl(i  /lens . 
In  an  I.  F.  OK  0.  I".   11  KHiNK  In   an  a.  I-'.   irK'HlN-K 

'.  i  Wi    =:    III    ~    /\ImJ, 


«1  _  t<l 
r,  ""  r-, 


Also,    (lisrfj,'arii;tij;     blade     thi(k-  Alsd.  (li-^nx'ar'linL;     IjI.kIc     thuk- 

ness,  i"'^-^. 

A^  =  irrt„i,  and  A,  —  znr^J,.  .L  =  :tA',/,   ami   ,/,  =  zirA't/j. 

Ut'l.xtion  bdU'fiii  tht-  lip-itiii,'U-K  /u  Lit  ion  hetzvfni  the  /.■/■-,i'/;,Vi. 

\\\  (11    I  C)  •""'  t''*^  tiiani;lf  .?<(/,  lU'  <'i|.  iV'!  ■ind  tlu'  iriaii^jlf  ,;../, 

I  ,',/, -in  r  _"'_■*'"  t"  +  >''       ,,-,  </,  sWi  ;- _  K,  _  sin  (.t  -)  ^, 

;,',/:  tan  /i  ^  <■,  ~        Sinn       '        •"  I       ,A  lan  /i  "  ;',              sin  tJ       '       ^' 


or 


'  or 


,,t /i  =:  cot  r  +  cot  .1.   .    (381  y  col /■<  -    c.tr  t^  eoi  .1.    (38) 

1.1  A'    I  li  IN      1  1K1:IM  s. 


In  an  i    F.  <ii-:  <  '•  I'-  1 1  KiiiNK, 
Spied  of  tiirhiiit. 

\\\  <.!<.  (J4).  (3;).  (37), 

y/Atot/» 

tan  /7  +  2-;  cot  ;■ 

1  'tlocity  of  ij/lu.v. 
V,*  =  w,'  tan'  /< 

y/A  Tan  ^ 


19) 


-•   .      Uo) 


In  an  a    !■■.   I  fUHiNK. 
Spct-(t  of'  tu>  I'iiu. 

i!y  (■(!>.  i:S),  (35),  137), 

:•<.''//,   +  //»  rot  /"^ 
'"■==  -  ,  ■     '39) 

tan   fS  +  2     cot  ,>' 

'  Vtloiity  of  <ll!i<x. 
V;'  =  «,» tan*  /<. 

2c(//,  +  A)  tan  /:f 


lau  n    <r   1  ^  cot  >' 


tan    p   ^  2  .   i-ot  ;' 


UOI 


THlAIRY   Oh    riKHINES. 


'^ 


.li/i.  lint  IJ  of  -valtr  pasiini^  Ihrnut^h  '  Amount  (J  <•/  -uutter  passitt^  throus^h 
till  turlir.i-  pel  s,-c,<>ui,  blade  thuk  ,  tlu-  tut  ''ine  /,  /  s/io>ui,  hiule  thiiic- 
Hiss  hetttf^  dhri-f^anieu.  iiess  htiHi^  i/isr,-^,i>  lici. 


-  2T>  jt. 


ta  1  f) 


Zi,//:  tan  ri 

:  ^   cut  y 


:  Q  =  2nA'i/,T,"  —  i^/i./jT'.. 


V    til"   fi    +    -','  C'lt 


(41' 


-/, 


'J7i,    u.ffu/  7i;>>  k    u//iri[i;,ir,iinj^'    liy-     Thf    ns,ful    'fork    \Uistei^ariitnj^   //>■- 
li' auin  >r:u'\ttitiiis) 

'V  -X 


'    ''' 
I    +  tan  /i  t.in  y 

Tht  wrttspofuiini;  ij/i.  ii'iiy 
I 


I   +  \,    ,   tan  /j  laii  ;' 


142) 


(43) 


iiranUc  >  esistttiufs) 

<.yQ{H,  +  A) 

T     f         ,    tan  fj  tan  y 

Thi'  iOtf e-s/wndiii^'  ijluu-iu-y 
I 


I    +       "T  tan  (i  tan  y 
-  tit 


(43) 


It    is 


sonicliiiics    assmiK'd,    Init,  It  is  somctinics  assumed,  t)iit.  f,'pn- 

),'cnt-ially  spcakinj;.  as  a  (^uidc  only,  orally  spt-akinj;,  as  a  .miid<-  only,  that 

tliat  the  inU-t-lip  is   railial,  Fi^'S,  299,  ,|,,.   inlct-li|i   is   vcrlual.   I'l^;. 

300,  so  that  that 


<r  =  '/j°  and  lu  -- 


,'.  301,  so 
(  =  90°  and   ?/,  =  i'u.'. 


?4,;  ; 


thr  elli 


lien  IV 


•     1441 


.ttm 


THh'nR)     <>/     TLKfi/XFS. 


3' J 


An    approxiin.iti'   e-tiinatc   of    lii'   i       An    api^rcixiin.iti'    r-tim.itc   of    the 
|peed   of   iht."    lurljinc    may    now  he  I  six'cd    of    tlic    turbiiu"    niav  now  be 
Dlameii    t)V    inakiiii;   the    ctFiclency  I  ouiained    1)V   ni,ikin|ij    the    efficiency 
(icrfcct,  wlicii  I  pt-rfect,  wIk-m 

III-  ~,^//  ■  ■     .      ■     (451  "r  ^j,'//.    r-  ii).       .     (4i) 

;   iqs.    (20).  (37).  139).    I'""   diff,-i ,r.ie\lly  f</i.    (20i,  (371.  1,39).  tkc  Uiff,>,-nie 


:t-t:ct<it  till'   iiilit  iiiui   outUl  pf,s- 
ure-luadi 


/.  -  h 


//. 


(       (kJ;      I 


2^ 

■idi  \ 


belwefii   tlie   inltt  ci>til  oiitUl  prei- 


I 


i        sin'ril-r2-''(otrc<)t/3)l  '       !  sin-rH-2yCotrcnt/ij  I 

I.  "1  .'I  "'  /  I 


If  tlie  turl)ine  is  above  tlie  surface  H'  the  turljiiic  is  al)(;ve  the  surface 
if  ihc  tail-water,  there  will  be  no  in-  '  of  tin-  tail-water,  tiiere  will  be  no  in- 
•i'uv  of  air  <lo"'  '  :  ''ir 

if  /.  >  /.•,   i.e.,  if 


if  />'.     ■  /-,    i.e.,   if 


-  n-,i-    I +:TCot  r  i^ot  /i:  ,■  ^-.        •  '  ,,       , 


sin' ,»'[  I    f  2'  '  cot  y  eot  fi 


If  the  turbine  is  drowned  with  a|  if  Uie  turbine  is  drowned  with  a 
iiead  //■  of  water  over  the  outlet,  i  he.id  ii'  of  water  over  the  outlet, 
there  will  tie   no   b.u  k-flow  of  water  1  there  will  be  no  baik-llow  of  water 

if  p\  ^  pi   ^    '•>/''.   I.e.,   if 


h7 


s.n.r^.   +2^^- 


,  cot  y  cot  (i 


if  p<   ■■  pi  -t-   '■"'/',    i.e.. 


C:)' 


sin'  y'\  +  2  r  cot  r  cot  IS 


')' 


Spri'ii  1"  /III  hi'ie. 
Hy  e(is.  CD,  (37). 


ri        r,  lix  sin  y     ,^-^yj  ,     ^  | 

Ui=       H:=  -  -J l/igH,.  (47)  I  »,    =    « 

>  I         V,  di  tan  fi  '    "  I 

V/,)<  I'tv  .if  ■iJbi  t . 


i.Mi'ri  ^1:  TURiii\i:s. 

Spc'i'd  of  turbine. 
By  eqs.  (21),  (37), 

1/1  sin  y 
.Atari  /i 

{  V/i'i  //i  ('/  ,illhx. 


i  -j.'''''i-     (47) 


„      /  I  , 

v,~UiUK\\(i^     ''  '  sin  rv  2i^//,.    (48)  li'i ---//j  tan  /^  - '-' sin  ^^  t  ^i--//,.    (48) 


5M 


rm-ORY  oi   riRiiiM-.s. 


///(    luthiiie  pi-r  srciiud,  f-laJr  tlinh-  \       th,-  tuihnir  po    MCn.i.  blade  thi    , 

ness  Irni^  di^t  i-i;a>  di\i.  >i,\,  /i,-/iii;  <!isi  ,xtt>  ii,d. 

(J  =  Z7rr,d,Tr'  =  2'r/-,d,7'i  sin  v       l         ij  =    •7rA'd,7>,'  —  znh'd.v,  Mri  > 


2!Tru/t  sill  }'  i  iii //u 


'4vt 


=  J,t/w/,  Mil  y  \  i.^/{,.     ,      ,4,y 


diatlll,    >  ,\i^t.l>U  is)  dfillllii    >  i-\/\l'tHii'^) 


S) 


-\ 

=  i.)l>//J,\   —     ','  ■   sill';/].      i;oi  I  =i.)'Ji//,  ^/.i[  I --,,',''  siii-'v 
\  '  .  ('.'  /  \  \       //i  -<-  lid r 


>:'dr  .  , 


rill'  ,iiii;-ipi'iiiiiiii:  ,p'ui,iu  1 . 
//.      d;'       „ 


An  t  \]ii  i.-^^ii  Jii  i,m  aUii  be  t.i-ily  i  ibt.inu'd  L^is'ini;  the  clVi- 
I  Hill  y  ir(|.  ;i  I  lit  till-  A.  I',  tiiihiiic  niilr|)cii(|t.-nt  nt  tli' 
1)1', 111,    // ,        I  lin-^,   liy  ci]--.      _'S,,     3;  I.   aiiil   147), 


m  ■ 


'> 


:lj»^ 


,/,  ^in  iv        (/,-  sin'   1-        //,  -4-  // 
,/,  "t.ui  H  '^  ,1}  tan-  li  //,      ■ 

It  in,i_\-  1)1  .i-.--nni{(!.  a--  .1  iirst  a]i])ii  i\iiiiatii  m,  that  in  im 
jiukc  turliint  ■-  the  wlmk-  nl  tin  wati'i'^  ciicit^y  at  inlet  i 
tiansfiirincil  into  iisclul  wnrls.        I  In'ii 


'  1 


"I'l  ^'''-^  Y 


'ihcrcftnc 


i\  —  li(^  cos  1-        2\\  cos  r. 


Sl-:r('\l).  (.'oiisuki  the  li\]ioth<-i-  "thatat  the  <  utlet  tlir 
relative  velueity  ul  the  w  ile-r  ami  the  ocnplieral  liiie.ir  \(.:!ocit\' 
of  the  \\heel  are  ecjual  ''       i  hen 


II 


^5^ 


THFORY   Ol-    riRMSF.S.  515 

'1  Ik-    tii;ini;U    ■  >t'  \  dc  nitic^,  ///,',    ,a    diitlct     i^    now    therefore 

an  isD^celes  tnanijle.   m  uliieli  //.•  —  lIi,  ami  the    ani;lt>    //fk  —  li 

l< 
—  >/:>    —  1  lierelorc 


■F 


Ill .  -'in  2  /  ,  sin  -  . 


153) 


Kq.   (5  i,  ai,Min,  ,^ivcs 


.-/,:■,  sin  )'  —  A.}'-,  sin  fi  -=^  ,  /  ./^  sin  fi. 


(54> 


7.F. 


A   "'   k 


\\',.   -so:. 


"  /■ 


Ki  .    304. 


i  ^     >  N    A.    I'.     I  I    klilNF 
«,    =  //j    =    jVi.'. 


Ill  ric>  ill   i  'I'lli.t  /jiiia. 

In  an   1.   I'.  '  <H  (  ).   1-".    1  I  ki:lNi 

"1  Hi 

r,       >-, 

Also.  (iisri'i;,ir(li!!!4  biadr  thirknc^s.         Aisr),  rlisrefjanhnt;  blade lliickiifss. 
.■1,  —  2T;  ,,^  .     ,-/,  _-.  :,T/ .,/..,  ./,    -  ;tAV/,  :     ./,  .=  zn/u/.. 

JkcUUioh  ikt'Vitii  111'-  lip  ,ii!i;i,\.  Kilalioii  I ,!:c,eii  t/:,-  lift  ant'li-^. 

H\  i-i).   154'   and   ilif   tnainjc  ./,  1/,         B\   i(|.  1341  .iiid   the  iii.in^lc  ,;.,/. 

I'l^s.  ;,u;.   i03,  Fij;.  304, 

>  V,   'iin  y       «,  _  sill  dr  t  y)  '      'A  sin  ^  ^  «,       sin  (•!    <    ; 

I  .'di  sin    ti'~  v,~        MM  .1       ■       3>  ,/..  sill  ii~  Vi  ~         siii  .^  "^' 

(If  '  or 

r  \i.  ^  ,  'A 

^  ,^^    cusei   /3  =  eoi  y   +  eut  .i.      i;(ii  !        -  losec  /J  _-  lot  v   4    ,  .,1  .,       ,  ;,,) 


S'6 


iHi-nRY  !)/■   ri:RHi\f:S. 


||5 


If:  " 


ii*j' 


•^ 


V' 

it 


■-^ 


:^i 


K  !     \  I     I    I  1  >  \      I   I    K  l;  1  \  K  s. 
In    \N  I    F   i«i;  ().  I',   i  I  Ki;i\K.        '      i\    \  n   A.   !■'     i  i  i;  m\k 
^/>V(/  .y  luibiit:-. 
I'.\    iqs.     J4),  (52). 

«,7'i   COS  >'    =^'//i. 


•57) 


ami   liiTi  ;.v  liv  I'q.   (  551, 


VfLuity  ■>/  rjiiti. 


4Wi    -iiir 


y< 


2^H,  -.  tan     tail  y 


^petii  ,'/  III) /'III,-. 
My  cqs.  (28).  (52). 

«,r',  ros  y  =  ,t;(//,  +  A).  .     (57) 
a  'd  lieiu c,  l)v  11).  1 55), 

,,  ,  ,,         ,  </i  tan  y 

I'tloLity  I'J  ■jltin. 


'59)         =  2^'(//,    +  /:/  '  tail  ^   tan  y.     (59) 


Quanltt^  O  ,.f  7i.it,,  fuiumx  lii>oii);li    o,„vil        J  ,•/  \„/r,  /<a^suif;  l/noui;/, 
l/u-  tutl'ine  pfr  snomi.  hhuic  liiuk-        ,/„■  i,„h,::,   p,-f   uYoii.i.  Made  th'uk- 


fj  =  zntjU.f."  =   2»»  j</,7',  cos  ; 


Q—  2nhd,vr"  =  2n/uir-  .  <<'S- 


=  2'"iV'<//i<'',</jSiii /3  t.iii  >-.  (fifj)        -2tA'4/^-(//, +//)./,. /i-iii/it.m/-.  (fio) 

y/it   if^iful  v;nk  (,iisr,-s,ndi,ij;  In-     ji,,-   i,„/i,l  -.t;>rk  {,iisr,-^ar,iiHfc    In 
liraulh  r,M,/,t'nfs)  lir.tuiic  f.sisr.tncfs) 

^        "■-  i  \  -.1;  / 

«7WJ//,^i   -|/|ti"i_,   t,.ii.'j.  Cl)    =«,rJ(//,+///i_|''tan'^t,ui>y 


(I'll) 


JAf  cortesponifini;  fffitieiu 

=  I 


.   tan      t.in  y 
lit  : 


(61) 


J  /if  i\>rreipomiiiiii  fjfiiuncy 

■/,       n 

=   I    -         tan       t.,11  y  (6j) 

III  1 


/."i    -,•     <2o).  (srt.  (58).  t/ie  iiifft-renct    liy  rq'..  (20),  (57*.   (58).  the  .iiffccMif 
htltii-iH   tlir    ml, I  ami  outlet  prti-        hthcfn   t/i,-  inlet  and  outltt  prts- 


iurt-lit.i,'^ 


/*.  -/. 


// 


\         r,V,  am  lyj 


(63) 


sutt.lieitd\ 

/».  -  p, 

■U' 

//, 

,,  ,,         ,  dt  sin  /9 

//,  -  t//,   -t-  /O  .       -— ,      163) 
1/,  sin  2y  ■' 


THEORY  OF   TURBINi:s. 


5 '7 


If  tlic  tiiit)me  IS  ;ibi)ve  thi'  suiface 
"il  I  he  Iciil-walcr,  t  liL'tc  will  In'  no  iii- 
llow  i)f  .iir 

if  /i  >  p,.   i.e.,   if 
sin  2K       r,Vi 

If  thi"  tuil,  •!•  »  ilrowiied  with  .i 
liiM'i    /'    Ml    w.iMr    mVit    the    outiel. 


If  thf  turtjinc  is  above  tlic  suffice 
of  the  l.iil-waliM,  tlu-if  will  In-  hm  iii. 
tiow  oj  ,,ir 

if  p^  >  pi.  I.e..  if 

sin  ly      IL  +  //  d, 

Ivnfi  ''       //,      ./,■ 

If  the  turbine  is  itiowned  with  :. 
Iieail    //■    ot    watej-    hvit   tin'    nut  let. 


IIktc  will  lie  nil  !...>  k-!l.iw  ..f  water     :  there  «  ill  lie  lu.  Ii.k  k-fl nv  ..(  w.iti 


if  p\  ^  p,    +    n'//'.   I.e..   if 

Hi  —  il'      r ^''di  sin  li 
H       '^ ri''dx  i,\n2y' 


it  /,   ,-  p,  -f   i.'i',  i.e..  It 

sin  -iy      Jl,   +  /i  ,/., 
sin  IS      Hi  —  h'  di 


IMIl   l.:-l      I  IKMM  >^. 

I.N    AN    1      !•'.  "K  ()     I   .    I  I    KHIN  h  In     \  N    a      I'      I  I    Kl'INf  . 

Hy  cq^.  (-';).  (5.M.  Hy  eq-  i,v>).  ij:,'. 

«■  =    /• (64J    2.47'  =  "»"  -    r,'  =  «,'        J','.     (64) 


Fi(.     305. 


\\'..   j(i6. 


Then  the  iiilel  tri.inijlc  of  veior- 
ities  .J../,  as  well  .IS  the  DUtlct  tri- 
ttii^le  fkh.  IX  also  an  liiosieles  tri- 
angle. Ki^s.  JO}.  ji)6,  unit 


1  l,itef..re 


<«  =  180'  -  rr    •  iftS' 


Ther(  (nre 


«,'  -  :VtrA  =   {'.• 


and 


fi 


..» 


Kifirour*  y +*4-x(//.+A).    (65> 


5' 


THIiORY    OF    rCh'HlXfiS. 


\l  '-i 


'^ 


IJ^ 


^H|^#' 

! 

Hv  cri.   (vp  ,inii   ilic   isosceles   tri- '       ,, 

_        _  d\  SIM  y       u,        //,  -*   li  see  ,»' 


</j  sill 


//, 


»  iVi  _   sill  /S 
/•jVj  ~  sin  ir 


(66) 


Spifd  if  luriiint. 


>•;  >  :     I', 

»j  =      «i  =       .        sec  >' 


I'elofily  of  fjflii.i 


T'j=  J«iSIIl 


/)■         '  . 


/  1  (III  } 


\    .V//..      !'■«»  I 


</.        //,  Sin  ri 

,i,  I/,  +  //       -in  zy 

Spi't-d  of  tut  bate. 

//, -r//  -<-i    y 
""  =  '"=      //,  .-     "■ 

//,  •  //  -CI    y 
//,         J 

I'eliHily  !>/  fjfliix . 

//,  1  //     n 


i6f)) 


t  :>,■//..       (^-7) 


>•»  .'c''''-    ''«) 


(Jniintil\  n,i/  ;v,ili-i  /••iM/Hf,  tliroiif^h  Quantity  (J  i'f  -.vatt-r  /•,iMi>i,r  i/,,„.,^/, 
tilt  tini'iitf  /^n  sfioiid.  />/,t,/,-  tin  i:  tlic  tiir/uii,'  /<•»  unmd,  hUnU  thttlc 
nes\  /vitig  ,//.(>, xiif, /fit.  j      "'O  '''•-'V  dnrixar  /,./. 

*)=  lJrr,d,;>,'  =  ziti  ,i/,f,  >iin  y  Q-2nl\'d,->,'  -  znNii  7'<  sin  >' 


—  i»;  .//isitiyV';^'//,,  df-v  ' 


-  :» Ai/i »in y  i  ^x'' •  ('»9' 


T"//,-   ii^f^iti    Ui'ti-    {,fts>yj;,i»,//>n-    liy.     I'l''   ustful    '.t'ork  {disrfi;,i> Ji>x  hy 
ihauli,  i;\i\t.nt,<:t)  d'aulic  tfuslaiicft) 

=  .■,('//(,        '  ')        .     f-oi       I'-'.l //>•//)( I-     ■,,    sin'^scc'^Wo) 

\         /,<()%>•/  \         •'  •  / 

/       /  ft 

=  :vQnL-''l  tan  r  t;m  '\     (jxy  ^»<J^"\  •' ih -'^'t;.M  .- un  J.  n,, 


I'h*  lOttfsfioMiiiiii;  tfficituty 


r/ 


sin' 


.     .     (7J) 


./.  ft 

I  -  ^^-  tun  y  tan  ^.      .     (731 


I'kt  iiirrfip!<ii,ti)it^  iffitiincy 


I  —      , ,       sm-      scl-  y      tri) 
11  i 


=  I   -    .  tan  y  t.m 


'731 


I'K.ICTK-  1/    rofi  IICII-S'TS. 


5'9 


5.  Remarks  on  the    Efficiency. —  1  In    (  xiMc-^inn-,  -i\-m^; 

:1k'  rffn.iinL'i-  in  tin    pn  ^  cilin-    .  Iciliu  tii  iiix    ,{\r    ail  miirix'lliii  ill 

■  if   the    head,   an.'    it    |m11ci>a-    thai    taihinc--    w  ■  irk    i''iuall_v  well 

.il)ii\f  a:1(1  \h-\i  i\\    w  atfi 

'/, 
1  hr  ittu  nih  \  .   a;^ani,   iiu  iia~.c^   a~    tin     latiti         (ImiMii^hcs, 

ft  . 

hut  It  --hiaiicl  111     uiiuiilhiri  il    that  lli.    \a.hu    .  •!'  ,/,  luiist  imi  \>r 

tod  ^mall,  ,1-  thi^  mr^lit  iaii>-i.-  a  1.  mmh  .u  ti- ui    at  mUimikc  .iiul  a 

V  ■  ini'-.])iiiulm;^    1ms.,    (i(    I  in'r;^\  .        Ihr    w  ln(l-|iass:i;^i,-'^    shdiiM 

ilwavs  I'lin  lull  Imhc  ,  ai'a!  thrrcfuri'  (/,  mast  nut  In-  tun  l,UL;f. 

l-'iti.illv,    till'    ctlKiLiKS     mcrca'^L's    a-    tin    aii;^k"'    />'    ami    ;' 

"iinimiNh 

6.  Practical  Values. —  The-  ri.ll,,\\in,^  an  the  values  whieli 
•  \]>i  ruiue  im  Ik  ales  a  -  .l;imiij4  i;nc)il  results  111  pra'  tii  e,  hut  they 
-ill  iilil  he  iinl\    feL;ar(leil  as  ;^uiiU'^  : 

1 .1  t  ,■  he  the  theoretieal  \e|iitit\- due  t.' the  In  ail  //,,  ■"  lli.it 
:■         2X//,  I  Inn 

/>/  (11/  /.    i.   litutii'ii  tiiyi'nh' 


7'        SS    t' 


)'  Usu.ilh    \aiie      liniii     \i<     t..     ;(i        ni    .ivera^e    \.ilue    hcin^ 


If  «j  —  \  ^.  t^  iisuall\  \aiies  fri.in  1  y^  to  I  -in  ,  111  ,i\eraj',e 
value  heiil^   145". 

i(    >  „.  O.   (i  usu,ill\   v.iiie-    from    30"  tti  4^   ,  .iii    .iveiai.;c 

value  lH.in(4  35  . 


5-°  iK.u.ii(,.-ti   i.oiihKjf srs. 

Ill  ail  O.  I.  It  act!,. n  tnrhtnc 


ii» 


.2\v  to  .177, 


I.ct  I!  hv  tlic  iiuinhcr  Mfthc  i;uuli--l)ladcs. 
i.rt  //,   l)c  tin    nnillli;  1    .^  the   w  lirrl-hl.idcs. 


8/, 


''.  —  ;•,  — 


4  X  shortest  ili-t.uu'c  liftwrrn  nlufl-liladc 


2r. 


—   =  sh,irtf-t  (li'-l.iiuc  tutuci'ii   -ni.lf-bl.iilcs, 


//,  [.)      ,■.-. 


'> 


■|lu'  II     \\  ^.  .i;;v//,'. 

;-   iisuall)-  varies   from  20'  to  50  .  an    .iv.ra-^'c   valiK    tuiiu 


2^ 


It,- 


0  \\^{\.\\\\-  \ari(-    Imni    jo'    to    \o\  an   a\ira-c  \aliic    iR'iii" 

/'/  an  .1.  /•'    r,at/iiiii  tmltni, 

."V    :^  rv     =  .15.-  to  .2:'. 
//,  =  /,,  -^  .  567.. 

r  iisiialU   \arii'-   Ironi    1;     to    ;o  ,  an    .i\iTaL;<    x.ilm    hcinL; 
/::<  \isiiall>  varies  trom    1  _'    •>  30",  an  avfra{,'e   xmIik    hrin;^ 

1-or  a  ilelivrry  of  \o  to  00  tu.  t't.  and  a  fall  of  25  to  40  ft 
)' =  15    to  iS       and      0=   iV  to  16'. 

I"i    I  ,1,  livtTV  ot  40  to  .'oo  on.  ft    and  a  fall  of  5  to  ^o  ft. 
y  =.   iS"  tf)  J4'      and      ft  —  16°  to  24  . 


im 


F\.-1Mri  FS. 


5" 


V<<r  a  (icli\'(Ty  of  nion    tli.ui   Joo  in.   ft.  ain!  for  f.ill-^  'iflo^s 
IIkui  al).  lilt   ;  nr  t,  ft. 

}'  ;     .'4     til   \i>        and      /i  '^-  J4'  tn  jS '. 
hciidtiiiL;  ♦  .  /,  -in   y  li\-  .  /  , 


A'  niav  var\-  fmn;  '  ./'  t<i  2.\    if  ./' 
■> 

•'  ;  ti.  './  if  .; 


J     'iC].     ft. 

J  -M].  ft .  A\\\\  ■ :  \(i  SI],  ft. 


"    "      •'       ■     .  r  til    .  /  if  .  r  •  200  >;<i.  ft. 

In  A  I-',  i.iif'iihi  turtiinc-  A'  i-  nt'tcn  niaih-  ti>  vary  frimi 
"  A     tn   J.J   . 

4 

In  rcntjiin  and  inijuiNc  tiirliiiu"'  tlic  hl.ulc  iIiiLkiU's^  v.iric^ 
from  \  to  ■};  in.  it  tin  hladcs  .m  nl  \\rcuii;lit  irmi.  .iiui  from  \  in 
\  in.  if  they  .in-  nf  ta-t  imn  1  lir  tip-  nf  la-t-im;-.  I)lad(-  arc 
ii>-iiall)'  tajKicii. 

ICx.  I.  .All  a.\ial-l)i)W  niijiul-i-  tiirliini'  pa-scs  170111.  ft.  ni  w.itii  |i(r 
scconii  iin<icr  the  he. id  <>(  S.O  ft.  over  the  inlet,  and  it  in.iy  In-  a'^>iiim'<l 
that  tlie  whole  (i(  lliis  hr.id  is  traiisfnrinid  iiiK.  useful  vvnrk.  'I'l'o  depth 
lit  ilii-  wheel  is  .9  ft.,  its  mean  di.imiti  1  is  84  tt..  .md  tlic  <)ulliil;|i  ni.ikcs 
an  allele  of  72  with  the  Vertical.  Tiir  tiiiliine  li.i';  i^z  i.;!iii|e-  ami  'h)  whcel- 
v.iiies.  all  the  vanes  hem^j  J  1  lim  k  The  mitlrt  velm  iiv  ,^  whirl  is 
nil.  Find  the  direclinn  of  inntion  of  the  water  .il  iiilet  the  >l.ipi-  ol  the 
•*.  heel  vane  at  inlet,  the  II.  I'.,  t lie  speed,  and  the  inlet  and  outlet  nriticc 
areas  anil  widl  lis 

FtrsI       Uisrenard  liyiliaiil'i    lesi-t.mics. 

t'l'         _-        »itv'         ".?'    cosy 
Then  =  8.0  =  — 

jn.l  -',   =   2U,  r-os  y     -  K  »  H /> 

J  t.4''xJ*i  ft.  per  see. 
Also.  I'l*  =  t'l'  +  »•,'  —  11, M,  eos  y  .-  /«,'  ^  ;<,'. 

Therefore     J',  =  «,  =  //,.  and  the  tri.mifle  <t.i/ls  isosceles,  so  ih.it 
■  I  1^  180  I 

■■*  .        ,r  ■■■«'       8.6 

"gain,       iTZ'  ~        ''""  '■'''<'"'"•'>'        I     '  ,,  ~   .  ,  ~   •''*'5' 


ih'  _ 


and 


the  eiricienev 
» 


V     y  5 

3K946  ft.  per  sec. 


*;i 


Hi 


522  !:\' -IMI'fl-S. 

'riicrcforp  !/,   -  w,  =r  ;•,  (ot  iS'   -  j',.^vS  ft    pet  sec, 

2U, 


and 

>■!  th.il 


set  y  —         —  1.90125, 


y  =  59  5- . 

Tl„.  Ml-.       ''-'^   '   '7°^  ^'■' 


55" 

!'!ic'  -peril  in  rc\i>iut  ii 'ti^  jht  inu' 
1 70  1 70 


hr  inict  area 


7', 


si;-  ;' 


i/.)5    -;    Klip,  1  ',!>. 

()0  X   25.358 

8.38  S.|.   ft. 


^  53.08. 


I  lie  Diitlct  .iri-a   —  =  -    =  22.4  sii.  ft. 

J'.         7.589  ' 


S.-iS        ,/    '  .T    .    ,S.4 


62 


f«i 


<:i>sor  ;'/■  5-'  -   ,    I  '  --I''  60   |6'  ^    ^  Y   -  20  ^  ;  v/), 
</.  =  .408  ft. 


22.4        ,/;      T    '    8.4  -         ((.sc(    i.S         -    ,/.    .     iS.vi'iS? 
'  24  \  '■ 

;"|'l  <r    ^  1.223  f>- 

.^(ti'iiii.      I.iki'  liii-  liyli.iulii    rcsistunci-s  into  ci)ii-.iiUT,iiii>ii. 

V  '       S  .,  ,        "  7s.       ;/,7'.  ros  r 
-      <8.^i)  -  =  ' . 

-.V       v  .*■■  >; 

I'Imiii.  ,tc  r  :«    CCS  >-    -  22.1181;  It.  [ir-r -^oc. 

I  hi-  It  i,iijl;ii-  ,J. ,/  .-.  I  hiii-l.irt'  isosi cles    and 
/  I  =  w    =  «... 
-'  ■  lli.it  i(  i=  (.So'   —  1  y. 

Also, 

!o''''  =  !>''"•<■>=  '■'  *  '-^''  "'  t  -'■  ;>•-';-. 

'I  li(ic(rirc'  ;<  "1  I    I   scr    >-    -    i  1    —   57.6 

i"i'l  «    -  i'>  3^5  ft   per  sec.  —  «,. 


'I'lid 


7'  22.1189 


.1 

llctl.r    I.,, 

I  111    -|ii-iii  11:  1 1'\  ( >luli<  Ills  |n-i  inm 


r  =  47'  21 . 

<r  =  |8o'   ^   2  r  -  R5    18. 
'11  •  ■   16.325 

T   X   84      ~  37'- 


".186. 
1  In-  rllii  ifiu  V      -  -     81116. 

9.5 


m 


Tii'-  111' 
Till-  ;nlet  area 
The  outlet  are 


I  \-iMri.r.s 

('2\    X   170   y  9.5 


5-M 


X  ,8046  ~   [47.676. 
70  ci»'>    47'   :;r 


550 

I  71 ) 

V,   sill   >    ""  22.]  1S9 

; "o  1 70 


=-    10  440  Sfl.  ft. 


I0.44'^ 


'^/  ;.x  ^.4 


«.  tan  iS 
6 


.  ^  32.03  .sq.  It. 


24 


MC  47    J I 


-4 


C  ■-<■(■  85'   18' 


ami  r/     r  .57  ft. 


') 


cosec  ilS' 


/ 


3:. 05  =       </,  ,  ,7    X    .S.4  —  ^.,,^,    ... 

=  </.    ^    i4..S5Sf<5, 
i.ricl  ,/.  =  :.i  37  ft. 

l.\.  :.  .\n  .\  I  .  reaction  tmliiiic  .if  7  ft,  mean  (iiarmtiT  [lasscs  i(;« 
'  II.  f:.  of  walei  ]..  r  mm  oixl  iiiiil.r  a  total  head  ot  13.1:  It  ,  the  (le|illi  ol  (lie 
uiicei  henii.;  1  tt.  .\t  mict  Ilic  lip  ,mi;ir  ( ,11  is  (^.  ami  at  ..iitl.'t  th.- 
pcripiicrai  and  r.l.itiv."  veioi  itu'>  ,ir.-  <'(|iial  (/.,  =  «.,  =  /<, )  ■riiewiilili 
"■  ''"■  »l"-'-l  '^  I  :t.  at  111.  t  .mil  i,:3  ft,  at  outlet.  Deteimine  the  di- 
reMioii  ami  iiiaunitiuli' .,1  the  veloiitv  ■  ■!  the  w  .it(  r  at  ntranre.  the  up 
iiiit;l<'  at  outlii,  the  speefl  in  ^evl)lllti.>^^  per  miiuite,  the  eltkiency  anil 
'he  II   I'      l)isre!,MMl  liMiranlir  resistances. 

lo   I  ill-  I  .ihilitii  ni  ol  continuity, 
■T     "  .   I  .  7',' 
isiul  I  tier<  lore  r  . 

■\\i.iin. 

<i  X    1;.;  -:.'        !«<4  _  ;■,   »  ^   „,'  -    I  ,■  _   i^.    . 

•"■  -"  ■         ^^''4       "1      ",-!-•   J    i  H     piT  see,    =   //,,, 

.V,  IJ   f', 

9 


198  =.    T  ,Jr..". 

I  It.  p<T  Sfc  ..  J'.'      .  7j  It.  per  -.ee 


Jv  '. 


<»)t    V    = 


tin  /*  — 


Tiiri  I  Ir^re 


V, 
T',' 


;oi )      ,  III  1 1      1 


■ .)    -  3  ■ 


I 


/  .  ~  «. 


i;»'3 
I 

(  I. So' 


1',  =  211,  sin 


I  he  I  fill  iriii  V  i-    I 
Th.    Ill', 


i   ;  uf>4.     .inil     ri     .  20°  5'. 

"  . '     -•)"  r-r. 

•41;-      1744  ^"     ■  -";  It    i>er  sec. 
I    —  ,c>(h>.S   —    9391. 
X   .93<,i   =   .'85.25. 


64  X   13. 

<>2\    X    13.5    X     I9S 

5So~ 


il> 


r( 


•;J 


.,   4 


!i 


[*  ii 


?-4 


r\-iMriFs. 


.,  ,  ''■O.  X    1 2  t   3 

Krv.  .liilioiw  per  mm.  = •     =  ;6  6<S 

T   >    7  3   ■     • 

r:\.    3.     To  ((instruct    an    <).  F.    tiirl)iric    fr.-ni    ihi-    f.illowinj,'    data 
tliL-  fall  {J/i)=-  s,  h.  .    llic    interior  diaim  :i-i  ( 2;  ,  1  -1    iN    't.  ,  ihi-  exterior 
•  liainoter  i2rA  -  2.4s   ft.  ,    O  =  ;,o  cii    ft.  per  .si-c..ncl;    y   _  yj    .   ihi;  clii- 
lU'iK  V  ();)  —  .Q.      Also.  ilisrei;arcl  liyfii.iiiiiL  rt-sislanco. 

l-'ir^t.      Take  7„,"  ^  o.      Then 


•9  =   > 


;/,T',  <i      3o» 


Thcrcfort 


''4x5'         32   X   5 
=  4  »  -   ft    ptr  >(-■(•., 
an<l  ..,,7     --  <)'•  \   V 

.\i,'.iin,  lii,  llic  (  ondition  of     .  mtiniiitv  (oq.  5), 

T     >      I.JiO     X     ,/,,-',    MM    3O'    =    30    -    T     X     2.45     ..     ,/.j;^j. 

'I'.ikirii,'  ,/    =  ,/, . 


and 


•  '1   =         ♦   -    11.  iiir  sec. 


Tiicr 


"1 


_'/>  %  ^   _  216 


4'* 


\  It  It.  prr  -irr., 


and 


2.45 
18 


"•     =    <-    V    ('     It      jllT    S,-C. 


or 

and 


Ilen.e         ""  '"    +    '^"'   _  "'  __  ''-  <  3    _  »  3         <        . 

sin  0  -  „,  -      ,^„|      -    V-  -t    ^  '"'•■. 

240 !  -'      '^      • 


"'   -    lot;*  f3'        inlft-ii],  aiij,'le. 


Al 


tan  fi  = 


4  ♦ 


-  •.i''^49. 


""^  '^-  ->    V  "  ...„i,.,.,,,,,„,j^lf. 

I)isre);,ir(l;n^;  llic  (hu  knc-s  .,f  the  vaiic«. 

" '■"■■■■  ="""=..sr,„-='i'.-p-3 ....... 

the  outlet  area  =  .-},  =    ^°  rr  J?-   =  5,30;  s,,    ft 


It  wsl 


lid. 


F.X.-1MRI  t-S. 

TliL'  riiiml)iT'of  ri'Vnliiti(,M>  per  HUM,  —    '"-'^       * 


5»5 


'U-52. 


TIltTCluIC 


viiul 


:«j  -iiii 


r;-  -  2U:r,„ 


U'M  the  triaiij,'le//{7/  is  isosceles. 


Agai 


«i7't   COS    JO'   — 

i6<j 


Therefore 


and 


"  1        4  t  J  It.  pet  ^ei.. 
)-o  2.45 


coscc 


_  ,,  -  45  "'  /■^           49      - 

i.S   2  2           iS  '  - 
Hy  the  coii'lition  of  1  ontiniiitv. 

_</,?',    MM     50      .     l.S     --    i^^     _,/„•>■■     X     2.45  =    "■/,?.,  COS           X     2 


45 


Taking;  .1      .  ,/, 


^  'V,  .^.8  f'j  cos' 


Hent 


320 


.97'.     :=    9.8    i'j  COS 
^,      _    '(.S    4    2  /^^       4 


2 


or 

anrl 


^  4320  ♦  3 
2401 


^^  3- "67, 


^        35     U'        oiitlel-tij)  .in^;Ic. 


Hoiu'o.  aKo 
t'l  ^  I4"'34  ft.  ,>er  sec,  u,-^  9,261  ft.  [kt  soc,  and  »,  =  6,7963  ft.  per  sec 


or 

or 
«iid 


7'.    2,45  J',     2.45 
324 


.  ■/>'  t  2  COS  '^ 


COS  30  +  cot  ..  SIM  vV  =  ,—;'  y    C'^f^'  ^^  ^  ■4'i399. 
2.4 5  X  49 

cot  I  I  Ho  -  .r)  -  .804, 

•  t  -   1 28    48    -  inlet-tip  angle. 


I  ■  r] 


^j(< 


I)isicf;_.irilii!i;  the  tin.  knes>  of  the  v.uic^. 

i'^  }o  54 ^j 


the  iiik't  aiiM       Z.I      ~ 


-','  -  7',  sm  30    =  v.H  ,  2  '•■'^    ,  =  4092  *q.  ft.. 


and 


-/,     := 


4.092 


■J",  It.  -  ,/, : 


the  outlet  area  = 


}o 


1  '■, 


fj  '   ^     ,_     •    1  ''ioiS  =  5.5^94  -^'1    It. 


l!ti 


1^  ■! 


It;':    ' 

IK. 


'^ 


I  111-  iiuniber  o|  .!-volutioiis  per  hum.  =  '  ' 


.T    X    I  ..S 


i'.X  4.  .An  1.  !•■  i,'a(  tioii  tinljiiic  .^l  _-4  in-^.  r.vu'rior  and  12  ins.  intorir.r 
diainctcr  pa^.M-s  400  Mallons  ot  uat(  r  pet  .end.  'I  !ic  iidct  and  ontlct 
oiilKc  an-.is  .,ii-  (.-(iiial  and  the  d.-pili  of  th,-  latl.  r  i.~  1.J3  t!  TIh-  i.;u:de- 
v.uif  li[j  h.is  a  slop,- of  r  ni  ;;  and  thr  mlt-t-lip  ,s  r.idial.  I  Jisrcgardmi; 
\aiie  tluckiics.s  and  hydranlu  i< -i-tanc  .'.s.  lind  '.In-  tol.il  lii-ad  over  the 
inlet  and  al-o  tl',-  r(li.  un,  \.  tl,,-  ,,.i[]ft  velo'itv  of  whirl  being  nil. 

i5y  tlic  londition  ,,{  ,  oniiiiiiitv. 

.17/  -  ./,r..-  ^  4.U)    :-  >\i  ~  64  =  .t,v,"  =     lA't. 
'I  iicrefoie 

1;'  ^  V,"  —  T'.,  :=  f)4  -r-  ,T  .  I  .  4  =  SA  ft.  per  sec. 


d  till    head  eipii\  aient  t. .    r'; 


'   '''  ^   ,„   -  {..]  =  '■".■/■' 'U  ft. 


55; 


«,    =   T',     cot  ^^   =    j?.,'    1=   40  -     fl. 


per  sec.  =  2«.,, 


and  till-  iiM  I  III  head  =      "  " 


,.-  ='4o.^,-  ^  3.'  _-.  5 r  121  ft. 


.N 


the  /,./,!/ hiMd     -   l.oV.f>04  +   5l..^,U-lo    -    5J,,S-i404  ft.. 


a;:d  till-  ellK  lencv  =    ''   '.'''""'  =  .g.S. 


-Mso,  tlie  --peed 
Tlle    II 


60   X 


j2.,S-i4o4 
peed  in  re\  oliii  lon^  perm, 11.  —  '     '  '   =  388.-6 


550  "    '>8  =^  370.98. 

tan /i  ^  ^"  =  S^^  ^  ^'A  -.4.     and     «,,    4S 


-^'Ma.. 


^I^JiilM 


V 


/  X.IM'/MS. 


5-7 


Ex.  5.  In  lh<-  prccc'iiin),'  c.x.iinijle  sh.)\v  li.nv  tli,>  results  will  be  mod- 
ified d,  iiisti-ad  cif  III!'  oiitkn  vclocitN  of  wliui  hcin-  ml.  llic  relative  and 
l«-ripheral  veloeitif-  ,it  outlet  are  ecjcal. 

As  ijefore, 

<'r'  =  Jv'  =  S,*;  ft.  per  sec, 

«<■     —  40, ',   ft.  per  ^ec.  =  2«,,  ==  2  ^2, 


1  Mc  s].r<-il  111  ri-volutii  HIS  |)ir  iniii.   =^  ■»    1 1 


A^ain, 
or 


38S.76. 


and//     ilii   A'/.// he.id,  =         r^   iililft. 


Als 


MM   /5 


Tile  elfi.  ieiK  V 


>,,     -  -4.      and      n 
4///'  sin' 


-3"  15  . 
II S  I  -  cos  /S( 


Ti      H.P.  = 


iii\   X   fi4   X    ii..S',47i 
55*J 


(JO,*)'    y    0835309 
32    X   51.S34710 

.'(7')         ;'!'>.  toy. 


979- 


Ex.6.  Av  irte.x  iiii|)ulse  turbinr.willioiit  Hiiide-vaiies  hut  witti  32  wliepl- 
vanes  of  5-iii  thickness,  has  .111  e.xtenor  diameter  of  2.C-25  ft.,  an  interior 
diameter  of  ;.  1  ft,,  and  passes  30  cii.  ft.  of  water  per  sert)iid  under  a  hea<l 
of  jfio  It  rile  watir  enters  at  an  anj;le  ^.f  30'  with  the  wheel's  periph- 
ery, and  llu'  lel.ilivi'  and  pr;  :,lu-r,il  vclot  ities  at  oiiilet  aie  equal.  The 
wheel  dr[ith  at  outlet  is  3  tiiii.  ^  the  depth  at  in'et.  .Allowance  is  made 
for  hydi.iiiiic  res.staii.es  l,y  takin;..;  .<j^  .,s  a  <  oetti.  len;  o!  velocity  at  inlet. 
and  b\  .uldiii;.;  10  pel  >  riit  t..  the  In  !■!  .  .|  iivaleiit  lo  the-  rrl.itive  velocitv 
at  'iLtlet. 


Also, 


»'i  --    V4  ♦  ''4.5''o    -   177.955  ft.  per  sec. 


10 


1 1 
10 


cr 


;  .'  = 


Thereiore 


"^  io\  2.6J5  /  250 

^350^    " 


sin  (.»   -I-   30) 
sin  30" 


5-'^ 


l:.\.-1MPll:S. 


%    ^ 


nl: 


m    IJl 


or 


.\t;aiM,  u, 


sin  in  +  30.  r.    ^  f    -^^^    ^  .4847.'. 

'(    +   30°  =   151",      iind      .1    -   121   . 

sm  I  (r  +   jo")  sin  20' 

bin  It  Mil  57 


—   100.(134  ft.  ]icr  sec, 
and  11,        i-'o  3072  ft.  jH-r  sec. 

■n  I  .  '"'    >'     l™>''U 

1  he  sprcM  111  r<\ '  iMi!  K  ins  iH'i   nun.  =  -  Ji  88 

li>   the  <;<  mill  tic  ill  of  c  out  innirv  . 
/l,Tr    ---   A,v,  Mn  .>'         30        ./..,'  =  .1-1',  sm  /j        .(i;/j  sm  /<. 
Tliereli.rc 

,/,  ■   7!    ..    2.(._-5   -  jj    V  '   =-  ..;,  :^  ±.   ,.,,^(.r  VJ'-^ 

'"        (  4f>  .-■  -  "77.955 


Also, 
o 
10 


.  1 8  I  n 


.  =  ■.5."<r'>4i  > 

1.  ft. 

3 

4«                   \ 

.; 

■)'3 


COSfC    /> 


)sec  IK 


or 
ami 


So.  50-2 

1.0^192  :=  co.src  (jy.^yij,  4    .372637)  =  (oscc  li   .    .')i)6637, 
cosei   fi    -   1.534.     or     fi   -  40'  41'. 
,  hiTcfore.  ,ils(),  0  —  (15    39.','. 


Aijain, 


'77-955  ^''^  j"    =  '54' ij  ft.  per  sec., 


ami 


7'..'"  —  «jfi   —  COS  z'^)  :t  ,S().;o72    ■    .241676  =   1')  47:',  ft.  per  sec. 

Hriuc 

the  HI,,  i.-ncv  -  l^'l:4    ■^ii4:ir3_^So.5o7^^x_nM7£_3 

32  X  560 


no4'.", 


17920 

The  n  P   =1^'-'   "■   3°  ^   5<'o 
S50 


-  .78. 
X   .7S  ==  1485.2. 


ifr 


DR.-U  f-riHl.s. 


529 


7.  Theory  of  the  Suction  i or  Drafo  Tube.--\'ortr\  and 

axial-fliiU  turbine-  -unirtiincs  lia\c  tluir  outK-t-nrifict.  s  oprniii^r 
into  a  suctiMii  m-  draU  1  tube  u  liicli  cxtiinl^  dowinvanU  and 
ili-cbari^cs  /u/r:.'  tlic  snrrai.c  (if  tlic  tail-uattT.  \\y  sulIi  an 
arranL;'nKnt  ilic  turliiiu:  can  bt;  jilaicd  at  aii}'  c<  in\cnicnt  lu'i'_;lit 
ab(i\c  t!u;  tail-water  and  thus  liccniU's  (.-a-il)-  accc-siblr.  uliilc 
,it  the  same  tinu  a  -lidrtcr  Icn-th  nf  shattinL,^  will  -uITkc.  'i  ho 
-iKti'in  tube  i-  u-ually  cxliiulrical  and  of  t  oii^tant  diainrtcr,  ^^o 
that  ilurc  i-  an  abrupt  i  h.m-r  i.f  -citinii  at  the  c  KitK  l-surt'acc 
(it  the  turbnu,  ]>ri  uhkiiiL;  a  i.  orre^pundini;  In-s  uf  er,erL,'y  i)\- 
eiidie-,  ete.  Ihi-  Iiks  nia\  be  prevented  b\-  '^ci  forillinL,''  the 
tube  at  the  upper  end  that  there  i-  nu  abru[it  t  hanL;c  of  section, 
and.  by  er.idu.iily  increa-ini;  the  diameter  dnw  n\^-ard>.  The 
co-t  i4  cnn-ti-netiiin  i-  ere. iter,  but  the  acticm  uf  the  tube  is 
much  nviur'iwd. 

Let   //    be  the  head  ab^ve  tile  inlet-orifice-  of  the  uhcid. 

Let  //  be  the  liead  between  tile  inlet-( irifices  and  the  sur- 
face of  the  tail-w  ater. 

Lc't   /.|  be  tile  hi--  nf  head  up  to  the  inlet--urface. 

Let  /.^  l)e  the  I0--  (it  he.ld  iietween  the  ulleel  and  the  tube- 
"Utlet. 

Let  r^  bi'  the  v-.-locity  of  di-t  liarLje  from  tile  mitk  t  at  bottom 
of  tube. 

Let  /'  be    the  atnin-pheric  pre--ure. 

linn,  as-umin-  that  tluie  i-  n^  -udden  ^  han-e  "f -ection 
at  the  <  iutict--urf  ice. 


and  therefore 
.^,  -  /': 


2^- 


=   If  -    '^J  '    -   ", -h   ",  +  /',  +  /',>. 


PK.iFj  rinrs. 


1  i»  :.f-^  ij 


wluTO    //        //■  4- //'    =   total    head    ,il)i>\r    t.iil-w  ator    -iiiruL', 
and  rr.  .  ,',   /.^.  I.,  arc  t'N|)rc->scil  in  the  lorius 

".'■,-.  /'.•■,'.  ".!,^  "4^. 

/'_,  .    ", .   /',,    "„  Ix'inq-  rmpiriv.il  iMcfficicnts. 
A'^ain.   il'.'  tiTiau  r  lie  id 


jiiii 


•1 


\'* 


i  ;i 


//, 


•V   ,   /.      A 


'^''   '"  ',',."'.■  -  ",—  ''.•,-  /',-,). 


and  is  riitirid)- indepciid.-nt  oltlu'  p.i-iti,.ii  ol'tlic  turhiiu;  in  tlic 


tU.)C. 


Alsii,  ,1  J^  i-,  ilu'  area  ■■!"  tlif  outK  t  tVoni  the  sHLtion-luhc. 


./, 


./,.-,  sin  r, 


sn   tliat    ,-•.  (.an  he  i\pirs-,i'd  m    tfrni-;  of  ;    .  am!  ii'  iici-  - 

is  aUt.  indci)cnilcnt  ot  tin   position  of  the  turbine  in  the  tube. 

Suppose  the  veloeitv  ol   llow  to  !je  so  small  that  :    .  ;  . ,  /. , 
nia\    1r-  eaih  taken  eipial  to  ml.      Tlien 

„    A.'-         '[■ 

and  siiue  the  niiiiiniuin  value  ot  /..  js  also  ////,  the  iiiaximum 
tlicoretical  height  of  tlie  wheel  above  the  tail-water  surface  is 
equal  tn  the  head  due  to  one  atinosj)liere.       Aj^aill, 


■'s  i//|        //^!//j  —  /  J  COS  li\  -f    '  '  . 


WnX 


./,f,  sin  )'  ^  Q  t-.  Jji^j  sin  rt'        ./,/'.  sin  //       ./,,••,; 
and  Iience.  l.ikin-' 


.C//        .■,:«,  Lfjs  ;'  ^     ^  n,^ .  u,  COS  Ii)  —  I,.'  ^- 


it  -* 


W: 


^r 


and  tlicrt  fiirc 
2  / 


/C).V.S7;'\   .7.V/)   Ml.CH.-IMC-fl    Fh'f-risr. 


_^     I/,  COSJ^-\-  ♦'/<,,  .  ;^_,  cos  (i 


-f^//+,./j  =  ,.^  + 


=  t-.'  +  2r^„..  . 


cos  ;/  4-  ♦  /<^  .  cos  ^ 


U  luT.'    A-  =    —  ^'  '  cos   ;-  -1-      ,    ;,.^  COS  /if^  . 

Hence  it  TmIImu-,  tli.it  ,-,  iiicrcriscs  witii  i,^,  i.e..  uitli  tlu.- 
speed  I't' the  tiirhine.  if 

A  suction-tiilK-  I.  not  nse.l  \utii  an  ontw  .ud-ll.,w  tuijjii.e. 
but  a  sinnl.ir  result  i.  obtained  'o\  addin-  a  -urroinulinj,'  .-ta- 
tionary  casin-  u,tl,  |,rIl-nio.itli  .„itlei  .\  .inniar  <hitn-or 
mij^^lit  be  added  uitli  eiiea  to  a  Join  a!  turbnie  workin^r  wiil,,„it 
a  siKtion-tulu-  helou  the  tail-water  The  theory  -I  the 
dit"fiis<ir  is  similar  t-  that  .if  the  ^iictioii-tiihe 

8.  Losses  and  Mechanical  Effect.-  I  he  I,....  s  nia>-  he 
eniiinci.'tril  as  loliuu-,; 

I.  1  h.  loss  (/.,)  of  head  in  the  cliatniel  In  win.  h  th.  water 
i-.  taken  {..  the  turbine. 


A 


./,  bein^'  tlw  coefHu  ient  of  friition  with  an  average  value  of 
.0067,  /  the  len-th  of  the  channel  of  approach,  w  its  mean 
!i>dranlic  depth,  and  r„the  mean  velocit\-  ni  the    ,  hannel 

/.,  IS  generally  inappreciable  in  the  case  of  turbines  oi  the 
inwaril-  and  a.vial-How  types.  .,s  they  are  usually  supplied  A.th 
water  from  a  I.irt,'e  reservoir  in  which  ;■„  is  sensibly  nil. 

If -i„  IS  the  sectional  area  of  the  supply-chaiinel,  then 

^K'o  -  0=  ./,•■,  sin  y. 


I    'h^ 


5.^^ 


A(.),s,s/:.s  .-/v/'  M/:(:H.is:-:.^r  ihiicT. 


'~-^';«  V       ./„       /    J..- 

jl.    'I'lic  Iiiss  i/.,i  III  lu'.ul  in  the  L;uiiU-[)assa;;os. 
1  l)i-~  1'  i--'.  is  ni.ulc  lip  lit : 

i(/i    1  I'.L-    loss    (Ilk'    t'l    n.-sistaiKi'    .It    the   rntr.uic  ■   'iit.)    the 
piss:iL;cs  ; 

i/m    The  loss  ihic  ti)  the    hiilion    httween    the  iliiul    .iiul  the 
fiNeii  1)1. ide^  ; 

u  I     1  he  Im-^s  (hie  to  the  ^  iir\'attire  ot' the  hhulcs ; 

((/(   TIk'  loss  ot  he. id  oil  le,i\in:_;  the  .^ukU  -passaL;(.'S. 

'1  hcsc  lour  losses  ni.iv    he  iiK'uded  in  the  c.\l)rc'ssi(_)U 


II'  ■;' 


M 


f.  lieiriL;  a  coeffieietit  \\lmh  h,e>  been  t'ound  to  \.u\-  t'lum  .025 
to  .J  ,nid  upwards.  ;\>)  avera!:;e  \.due  of/',  is  .125.  hut  this  is 
somcuh.it  liiL;h  lor  l;oo(1  turhiiie-.. 

N'oir.  —  In  niif'iilst-  tiirhincsy,  has  lueii  louiid  to  var\  troiii 
.11  t..  .17. 

Ill       1  lu    loss  I  /,.J  due  to  shoek  .'t  eiiti.mee  mto  the  w  iieel. 
.,   Ill   order   that   there   111. i\    In    tio   shock 
,''  i\      at   entrance,    the    relative    \elo(;t\'    y\'\ 
yj   i     niiist    l.f    tani;etitial    t<>    tlie    lip    ot    the 
^>|   I     v.ine.      I'i'r  .iii\  otlu  r  \tloeity  (;•,    = /r<') 
,^    "' '     and   direction     (i,n    —  y\  of  tlu    \^.lt^,■r 


'I  H 


Fic.  307. 


A,  —  tlic  loss  ot  hi.i.l 


at  (  nti.ii.i  e.  ev  idciUly 

o,    ,  o   ei-  -f  (O')* 


~m  y   —  T",  sin  1')         ( ;■  ens  v   —  r,  ens  ?-)- 


(*■'  sin  y'  —  /',  "in  *»r'       <  .•  cos  y 


/  '1  ens  af 


LOSSBS  A\D   MFCHAMC.41.   l-FFI:CT. 


533 


(ioncrally  lo  is  small,  .imi  /.,,  is  always  ni'  wlicn  the  turbine 
.    uorkinj^''  at  lull  pressure  and  at  the  nnrnial  sjieed. 

This  h)ss  dT  head  in  shock  caused,  \;\  al)ruj)t  chaiij^'cs  nf 
section,  antl  ai<o  at  ,in  an^le,  ma_\-  lie  ii\oidetl  b\'  causin;.;  the 
''CCtioii  to  \ai\  i;i.'duall_\-.  and  b\-  sutistitutinu;  a  lontitiuous 
cur\e  lor  the  an;^'le. 

I\'.  1  he  loss  (/.J  of  licad  due  to  friction,  etc.,  in  passinj^ 
throu^^h  the  wheel-passages,  including;  the  loss  (hic  to  leakni^e 
in  the  space  betweei.  tlie  L;uides  and  the  inlet-surface.  This 
loss  may  be  expressed  in  i!ie  firm 


♦        ■    '  2-    "''./,  sin    lil    2.J.-' 


where  f\  \aries  from  .  lO  to    zo. 

N<||  1  ,  1  he  loss  ol  lu'ad  due  to  skin-friction  often  L;o\-erns 
the  dimensions  of  a  turbiiu',  and  render-  it  advisable,  in  the 
case  of  hi^h  falls,  to  employ  small  hi-h-spced  turbines 

\'.  The  !(-s  of  hcul  /..I  due  to  tin-  abrupt  (.liani;e  of  sec- 
tion between  the  outiet-surfaci'  and  the  suction-tubi\ 

.\s  in  ill,  :\  (=_///)  is  sutidenly  i  haiv^ed  into  ;',(_///), 
and  the  lo-s  of  head  is 


L.- 


{/i/i'f  _  [/ixy  +  i/ixy_  {hx)i 


since  //.I   i,  \cry  sm.dl   and  m,i\   be   disre- 
{^r.irded.       1  hus 


Fig.  308. 


_  (/'.sin  li  ^  r/i' 


.-.,  bem.L;  the  component  of  .''^  (//f  }  m  the  ilirection  of  the  avis 
of  tlie  suction-tiibo. 

ll  tin -re  is  no  abi  upt  ^  han;;e  of'  sec  t  ion  Im  t  \\  ci  n  the  out  let - 
suifu      and  the  tube,  A.  is  ml. 

^  1  rile  loss  of  liead  i /.^ )  ihic  to  frii.tion  in  the  suclion- 
tube       Assume  that  the  velocity  ;•,  o(  flow  in  the  tube  is  e<iii;il 


534 


LOSSHS  ^l\l)   MHCH.-IMCU.   IUFZT. 


fe 


iiii 


:i'  i':\ 


t  P 


'> 


w 


to  T',',  tile    \(MiHit\-  with  whii'i    the   \\,i:'r   Icavis   the  turbine. 
Also  let  ./.  be  the  sectiuiial  area  i-\  \^^v  tul)e,      'I'lKti 

/iii  =  /\'  bein-  tlie  cuefficicnt  <if  friction  with  an  averat;e  vali. 
of  .000;,  /   the  len;^'th  of  tlie  tube,  and  w    its   mean  hydraulic 
(lei)th. 

\  11,    '1  lie    Io<-,    (/._)   uf  iu:ad   li.io   to  entrance  si     ce    at 

base  ot  tube.       I'his  loss  n)ay  be  expressed  in  t!ie  lorin 


/,  sin  r  -,  ,-• 

. ;      ^  2.'- 


the  avcrajje  value  of  /'.  bcinj;  about  .03. 

\  ill  1  lie  lo-  {/.j  of  head  due  to  the  ener^jy  carried  aua\' 
In  ilu-  uater  on  leavintj  the  suction-tube. 

and  ,•  ,  usuail\-  \aries  from  I  »  2_^// to  ?  \f2i;/f. 

in  o,„„l  turbines  the  loss  should  not  exceed  6  ])er  cent. 
It  iniL;ht  be  reduced  to  3  per  cent,  or  ev(  n  to  1  p,  r  cent,  but 
this  would  l.irijeh-  increase  the  skin-friction. 

l.\.  The  loss  of  head  lA,^)  produced  b)-  the  hiCtion  of  the 
bcarint,'s. 

Z.g  =  nW    II ^ , 

/<  bcinfj  the  coefficient  of  journal  friction,  /I'thr  \vei.:.,l)t  o,  the 
turbine  and  of  the  water  it  contains,  an  1  /j  the  railius  ot  the 
journal. 

Ilcnce  tile  tot.il  loss  of  iicad 

-/•,  +  ^-f-/-.  +  /..  +  /..,  +  /.,.  +  /.,  +  /^,  +  /-»  =  i:. 

and  the  total  iiui/ianiiiil  itfiit 


=  '<l.-')- 


t.  % 


mm 


E.\AMri  F. 


535 


XOTK. — If  tluTC  i-  nn  ■luetic  Ill-tube.  /,,  =  0=  /-,;  =  /-. 
—  /,,,  ami  till-  tiital  lo^s  bceonus 

7'.,'         ,  fall    from    outlct-surfaco    lo 
/.,  -f  Z.,  +  A3  --A,  —  /'cj  +  ^^,  +  )       tail-water  surface. 

I'.  \  AMI'I.K. — A  viirtcx  turhiiu'.  \\itli  a  (Ir.ut-tulu  ol  the 
same  sectional  area  as  that  of  the  outKt-oririce  c,).  iiitiL;-.  passes 
icxi  cu.  ft.  of  water  ])er  second  under  the  head  1  1  (j^  tt.  'I  he 
<\terior  and  interior  diameters  are  in  the  ratio  of  5  to  4.  and 
the  outle  -  and  inlet-areas  are  in  iheiatio  .  .t  (y  to  m.  'I  he- 
direction  of  the  water  at  llie  ink',  and  the  outh  t  lip  aa^le  arc 
^nven  1)\  sin  v  =^  .25  -  sin  /.'  The  water  leaves  the  tube 
throuL,di  a  sluice  having  a  sectional  area  10  i)er  cent  j^reater 
than  that  e.f  the  outlet-orifice  area.  Tlu  outlet  velocity  of  whirl 
is  nil,  i.e.,  rf  =90\ 

Disret,Mrd  the  losses  /.,.  /.,.  /-., .  /•„ .  and  /..,. 


The  loss  of  he. id  t'  ■  inlet 
•  <        •'     "      ••      in  \ch(  ei-passapes 


=4- 

I'i 

r:^ 


•  it  shiite-entiance  ^/* 

carried  aw;.N   b\-  \Kater   ~ 


Hence  the  tot.d  loss  ,,t  head 

Hut.   b\'  tie.-  lotiditi'in  of  i  ontinuits'. 

.  J ,.- ,  sui  1=0=  A^l\  sin  /tf  =  A,v.. 

Therefore 


!•         .1 ,     sin  /)  7',  .  , 


l\       A^     »iny 


y.~  K 


..5     t 
I.I    4.4" 


'•HI 


53^'  HXAMPl.F.. 

Hciicc  tlu^  total  los'^  of  hc;id 

tilkill-    f,         .!,/■,_.  \2(k    ;il),l    /.  -_  .03. 

.\-,lill.  tin-    ■•  uytt'ul  "    lu-.ul 


ipi 


ii  H' 


V% 


"1'"',..         I 

5 


1  lle]Ltoft 


32    ■  4 


a ,  .  r   COS  y 


r.      -in  a  Cos  ;'         r,-      1  ^5 


-i, 


Mil  ^' 


■   6. 


.109375  ft- 


r;^ 


r.-' 


9-5=  2;'--''-   -'-"194)        ,-'  X:r.3f'94. 


2^ 


or 


2^  —  4  tt  .  .qiprov.,      an.!      J'.        \G  ft.  pcrscc. 
'1  he  hsiJkI  -work  i>cr  II).  cii  w.itcr 

'  ^5 

■-■-:  4  X  ,     -  s.4s;>  a  -ihs. 

I  he  ^K'urk'  ii'iisiiiiu,/  u\  h)\li-.uili(.  rcsistaiu  I's  j)cr  lb. 

=  4  X  .2'!         I  .04  ft. -lbs. 

11k-  ti>l(tl  wi'rJ  ])(■!  II)    .it  w.itcr   =  9.4775. 

The  ,y/?<-//7/o' =^  "''■'*''''-'  =   .So 
9  4775 

""•  111'.=  -i ^-  X  .89  .=  96. 1. 


550 


too 


400  l 

■■'.■=    ,•   cosecA/=r-^^  =^  -'5  s,i.  It    ami  J,  _     -^27.7Ss,|    ft 


tiX/IMPLli. 
''i  =  ■9^\  ~  14-4  ft-  per  s,-c  , 


537 


1/  I  i 


u,  ^l\LO.(i^  16V  ^^^  ^   ,  3.^,^,  ,-t.  per  sec. 


and 


"i  ~  J'2~  ".'  ;?'''5  ft-  IH-r  sec. 


A  1m 


or 


U^  _  sin  (rf  -f-  y) 
'\  --in  (T 


—  CdS  y  -|-  cnt 


t-  Crit  ,1-  sm   J/, 


Cota  =  _"'   cosccr  -o,t   r   ^'^'-^^^S 


--    X4-3«73-   '   5061. 


«  =  33"35'- 

If  th<-  ,!iam.,-t.r  ..f  fl„-  tiil,,'  ,s  e,],!,,!  Im  that  ,,f  thr  ontl.t- 
siirtacc,  Ni/..  4  It.,  ,n„l  it^  ,t.  ln^,,,-  ,-,V^^  ,s  n.,in,l..ci  so  tl,at 
./.  —  O.  tlicii 

(■ncri;\-  |HT  II),   ..f  \\,iti-r  cai'-icd  awav  =  '•'' 


"^ck    =    ^,v""-'^..6'-..,6i35i)-4x.o,;4 
1  Ilf  loss  ill  sli.H  k  lit  (Irart-liihc 


<■*<. 


TIhis  tlic  t<.tal  |i.-.s  .1..U   l)ci(>int.-s 


'V 


V} 


,^(.081  -u  . ,  26  -f-  .05448  -f-  .04705)  =  ^  X  .2695. 

/  * ;' 
As  hcforc,  tlu'  tmi'fii;  IrmJ  —     >' 


2^ 


X   2    1(194. 


Uu 


p 


u 


?3« 


HXAMri.R. 


17 


Ihcrcforc  tlic   total   head     —  — ^  X  ~  :-,7^9< 


atul  tlu-  cffiLJcni.)' 
Also. 


2.[0(J4 

2.3789 


=.  .886. 


9-5  =  ^',.y--?,7^'J'  <-^r    ^'^      3.<j(j3t't..anu/',:^  15.98711.  per  sec. 


It    thcii'    is    no    ilratt-tuhc.        "'      must    be    substituted    tor 

-.C 

'    -.C  2^'  2_f  16 

'llius  tiu'  tot.d  loss  iif'luMd  is  iiiiu 


J',- 


r/ 


■  ( .08 1  +  .  1 2«)  4-  .062  O  =        X  .260: 


/■■ 


uhicil  exceeds  the  loss  ( it"  head  .■.7/'//  adrati-tube  1)_\'         x     (>0()' 

=  .038    tt..   which  is    less    th.iii    tour  lumdredtiis  .  pf   a   tool   .uk 
is  priittically  inappreciable. 


«W 


/  x.-im/'ij:s. 


539 


EXAMf^LES. 


I.  A  rlownward-tl.nv  turhiiir  . ,!  241ns,  intfrn.il  ,l,amrtfr  passes  10 
ciu  ft.  of  water  per  secuii,!  under  a  head  ,,f  31  ft.;  the  depth  n|  the  wheel 
IS  I  ft.  and  its  width  6  ins.  Find  the  ethrienry,  assuinin-  th.-  whirlin.- 
vel.K-itv  at  outlet  t(i  he  ml.  j^^^  " 

..  .A  downw.ird-llow  tuihn.e  ..j  ,  ;t.  external  diameter  p,,s,e's  ' Zo  ,'n 
ft,  .,f  water  per  ,se,  und  ui  !e,  .1  h,  ,..1  ,.'.  4  it  ^  ihe  ,lepth  ■.:  tiie  wheel 
hein-  I  ft.  The  water  enters  tlie  whe,-l  ,,t  an  an-h-  .at  (,o'  with  the  ver- 
ticil, the  reee;vini;-hp.,f  th,-  whe.'l-v.ines  ,.  verticil,  and  the  vel<;eitv  of 
whirl  at  outlet  is  nil.  I'lnd  ,1,,.  mtern.d  di.inieler  ..nd  the  s.eed  in  rev- 
olutions [ler  luiniite,  i,^^     4'if!       A(,-- 

3.  .\  duwiiu-ard-flow  turhine  has  .m  intern,,!  di,, meter  nf  24  ins  'the 
breadth  of  the  wheel  ,s  (,  ,ns.  ;  the  tiirl.,;:e  pa.sses  3}  eii.  It.  pe,  serond 
under  an  elTective  head  of  if,  f-.  ,\ssuin:n,L,'  the  whirlinfr  velocitv  at  out- 
let to  be  ml,  lind  ihe  elheieni  y  and  power  of  the  turbine.  If  the  vane- 
lip  at  inlet  is  vertiial,  find  the  direction  of  the  vane  at  outlet,  and' the 
speed  iif  the  turbine  in  revolutions  per  tniniite. 

.?«?.  .93 1  ;    53.865    UV.fi=:y-rr2<'2'.     K/,.;. 

.1.    Disenss  the  preceding,'  e.xaiuple  011  the  assumpth.n   that    tlie'pc- 
ipher.ii   speed   ,.;   outlet  1//,:  is  equal   to   the   sp,  ,-,1   „f    the  w.tt,  r  ,,t   tli.il 
[I'lirit  r  1  l,ili\ely  tc}  tin'  whi-el  '  /',). 

•  /«>.  .v:.S.  55-ri5  ".!'..  /i  .-  .-,  47' a>i<l  /  -  Jo  jr. 
5.  .All  .ixi.d-llov  inipulse  turbine  ot  5  ft.  me.in  diameter  passes  1 -o 
cu.  ft.  of  water  per  second  i:nder  an  ellectue  h,-,id  .,f  ,S.o  It  ,  ti,r  dei.th 
of  the  wh.'el  IS  ,-,  ft.  At  what  anf,de  slimild  tm-  w.iter  enter  the  wheel  |.. 
niveau  eirieienrvof  .V,  per  ,-.-nt,  the  width  of  the  wheel  ln-,>i^  ,  onst,int 
and  disri'!,;,[rdiii^;  hydtaulii:  resistanci's  -     ?•„,"      o.  -t'ts.  ~   27"  iW 

Also  lind  f,,)  the  veloeitv  with  which  the  w.iter  .riters  the  wheel; 
'/')  the  speed  of  the  turbine  in  revolutions  per  minute.  (.  1  the  directions 
of  the  v.ine-edyc-  at  inlet  anri  outlet;  (,/)  the  velocitv  of  the  w.iter  as  it 
le.ives  ;h,'  uheel  ;  (,•)  the  pow-r  of  the  turbine. 

.I'H.    i,,i      2346    ft.     |)er    secoml:    ,/.,    43.08;     .,  )    ,,   -_,   ,30"    lo'- 

/J  =  42    lo';  '-.'i  10.74S  ft.  per  s,.,  ,,n.|  ;  (,-i  t48.63  II.  P. 

If,  instead  ol  assuming  that  the  whirlin-  '.elocitv  at   e.xit   is  nil    it  is 

assuii,ed  that  the  peripheral  spee-l  («,)  of  th,-  uh,-,-l  at  the  meau  radius 

IS  equal  to  the  relative  velocitv  '  V.)  of  the  water  at  exit,  sh.  w  how  the 

several  results  are  aflected. 

Alls,   r  =  2^'  H  .    (a)   23.46    ft,    p,.,    s,-,,.,ul      ,M    54.638;    ,,) 
■  f  =  i::4'  49'.  fi  -  44"  f'':  <./>  ro.748  ft.  pei  sec.n.l  ,  1^    148  6;   li  I'. 


540 


i.\.iMrii-.s. 


II 


,m  ■ 


AisM  sImw  liMW  the  rp^.ults  iiic  ,ilfi-(  tcil  uln-ii  it  is  assiiiiioil  that  the 
hvilraulic  rcsi^taiici-s  iitccssitali'  an  ini tlmsi-  of  i^J  per  cent  in  the  head 
eijuival'-'iit  t"  the  vel'icity  with  which  tlie  uati-r  ciiti-rs  tl;e  wlicel.  anri  an 
increase  of  lo  lerieiit  in  tlie  lica<l  <(|tiiv,ilcrit  to  tlic  relative  velocitv 
(  I'-,)  ut  outlet. 

^Ih\.    Wlicn   :■-,."  =  o  {.v,   22. u   ft,    ]rci    sec  on-!.    .    ,    44.21.    (, 
<r -:  14-    50'.  /i  =  27' 44;    !</)  10  ;4,S  ft.  jjcr  .MC.nil  .    .  .  14;;.(,3  H.l'. 
Wliiri  ;(,  -     r.j  (,(l  2.M  iv  It.    )er  se<  .>nd  ;  ((4)  50,97;   (()((-  123"  ly', 
ti       47    -^' \   !''■'  10.74X11.   per  sec  .11(1  ;   {y)  14.S.(.;  II. I>. 
If   the   turbine   lia>  ()■■,   ^'iiide-ljl.nlt  s  of  .  2-in.  th  lcklle^-■   and  (.3  uheel- 
vaiies  oi  ,4-111.  thi.  k'less,  find  tlie  wnh  h^  of  the  inlet  and  outlet  openings 
-•'«>     If.-:,"         o.     ,/,  --  4.214  ft.,  ,/,  =  2  83  It. 
If  II,     ---    I':  ,  ,/,         i,7.S  ft.,     ,/j  =  1.48  ft. 
(>.   The   ein(  lency  of  an   a.\ial  flow   turbine   of  4   ft.  nvin   diameter  is 
|)o  |ii-r  (  eilt.  ,ind  it   p.isses  12   cu.  ft.  jn-r   second  under  an    elle'Ctive  he. el 
of  40  ft.      At  the  mem  r.idius  the  w.iter  enters  ,it  an   pngle  of  30    with 
tlie  wl-.eci'^  f.icc,  and  the  uIi'iIiml;  vcI-k  i!\   ,1;  outlet  is   nil.      Find  (,;)  the 
velocity   wr.h   wliich   the    water    enters   and    leaves  the   wheel  ;  (/)   the 
directions  of  the  vane  at  inlet  and  outlet  ;  (i)  the  sectional  areas  of  the 
inlet-  and  outlet-01  ilires  ;  (,/i  the   speed    of  the  uheel  in  n-voliit;ons  per 
minute;  (f)  the  power  of  the  turbine. 

.his.  (,/)   32    ft.   per   second,    1''   ft.   per   second,     i/i)  <r  —  49' 6', 

■'i-Q\'  V;  '.i.r^^ip  ft.;  (,/i  i'):\i<i.  !<■)  49,',  H.l'. 

7,  .\n  ,i.xiai  How  turbine  of  5  ft.  nie.in  radius  passes  ?I2  cu.  ft.of 
water  per  second  utider  a  total  ellective  head  of  12. i  ;t.  .\t  the  niiMn 
radius,  .he  direction  of  the  inflowing  water  m.ikes  an  anj.;le  of  -o  with 
•.he  verticil,  and  the  \,ine-lip  at  the  .intlet  makes  an  anj.;le  of  17" 
with  the  wheel's  peii[)herv.  If  the  whirhni;  veloi  itv  ;!t  t!ie  oiitl.-t-siir- 
f.ire  is  nil.  find  (./jilie  velocit\-  uitli  whcii  the  u.iter  must  c-nter  the 
wdieel  to  j.;ive  an  elh(  lency  ol  .v;,i  per  cent.  .-Ms.,  find  (/')  tlie  drectloii 
of  the  v.me-Iip  .it  out  a  t  ,  (,  1  t  he  sp-c  (I  of  the  wheel  in  revolutions  per 
niinute  ;  1,/)  the  widilis  .md  are.is  (f  the  mlet- .ind  outlet-oril'u  e~  dr) 
the  power  of  the  turbine. 

.his.  (,n  i').9  ft.  per  second  ;  i/n  0  -  Sr  25';  ir)  37.67  ;  (,^)  .991   ft., 
31,148  sq,  ft.,  1.81  ft.,  35.14  s().  ft.;  bi  277.709 

If  the  turbiix;  has  41  i;mde-bla<les  and  40  wlieel-vanes.  ail  of  .25  in. 
thickness,  find  the  widths  of  tiie  inlet-  .aiid  outlet-openings. 

.Ins.    1.2^  ft,;    1.37  ft. 

8.  \Vriled(;Wiilhei  (juationsfor  iouval's  niodilication  of  l^ulersturbinc. 

<).  ,\n  a.Kial-flow  impulse  turbine  (.asses  170  1  u.  ft.of  w.iter  per  second 
under  an  elTe( live  head  of  9,5  ft.,  the  depth  <.f  the  whet  1  being  ,9  ft.  and 
its  me  in  radius  4.2  ft,  Tlu'  v.me-li;)  at  the  outlet  makes  an  angle  of  72' 
with  the  veitical.  Assuming  th.it  th"  whole  of  the  eifeclive  head  is 
trail -formed  intouselul  work,  and  licit  tlie  whirling  velocity  at  the  outlet - 
surface  Is  nil,  find  U:]  the  inclination  i..  the  horizontal  of  the  uutlct-Iip  of 


EXAMPLES. 


541 


the  i^uidi-vaiie  ;  (f>)  tlie  direction  of  llie  inK't-li|i  (if  ihc  whccl-vane  ;  (i  i  \.\u- 
cliu  irni  y  ,  /irsl  nc^li-ctinf;  liyiir.iulic  ri-sistanccs.  .i;i(l  .uwn.i  taking  thiM 
rc-.i^laiii'cs  into  a("(i>iiiit 

Am.    First.       {a)  y)"  5J  .   (/')  bo'  10';  (ci  .903 
S.iond.  (in  47'  21'  ;  (b)  85"  ij>';  (<■)  .S04. 

10.  In  the  preccditit;  c-xanipli'  tnid  the  mUt- and  outUt-orilii  c  ari'a^ 
in  the  two  cases. 

An-.    I'lr.^t.  8.3S  S(|,  fr.  ;  2:4     -.).  ft. 

Second.     10.45  si),  ft.  ;  32.08  sq.  it 
If  there  are  62  wheels  and   (b  L;iiidf-vaiie^.  thr  tlinkiiess  of  the  hitter 
ticiiij,'  .2  ill.  and  of  the  foimer  .4  m..  liiid  the  widtli  of  the  iiilet-oritices. 
.Im.    Fiist.   .40911.;  1.26  ft.      Second.    .508  It.;  1.81  ft. 

11.  .\n  axial-tlow  tiiil)ine  pasM-s  200111.  ft.  u\  water  pc:  second  under 
a  lieail  of  14  It.,  the  depth  of  the  wiiecl  htrini;  I  It.  Hie  mean  raiii  is  of 
the  wheel  is  3  ft  ;  the  areas  of  tlie  inlet-  and  outlet-surfaces  an-  m  the 
t.'tio  of  7  to  8  ;  the  w.iter  enters  the  wheel  at  an  angle  ui  21  to  the 
wheel  face,  and  the  outlet  edge  nf  tiie  v.iru-  makes  an  angle  of  I'c  witii 
the  face.  Find  the  speed,  elliciem  y,  and  power  (jI  ihi-  tiirlnne,  and  al-o 
the  direction  of  the  inlet-lip  of  the  vanes,  ■:■-,"  —  o. 

Am.    "3.69     revolution^     per     minute;       734;     ',23. .-'4-,    II. 1'   . 

"  -65°  57'- 

12.  A  dovvnw.ird-ll  .w  lurhiiie  of  3,",  ft.  nie.m  diameter  and  of  tiie 
impulse  tvpe  i->  supplied  with  5;  cu.  ft.  ■;!  w.iter  per  second  uniler  ,1  licafl 
of  400  It.  and  m.ikes  500  revi  iliitn  nis  per  minute.  The  water  enter-  tiu- 
wheel  ,it  an  ,iiii;le  nl  sill  '  .6  wilti  tlie  horizontal,  aiul  the  depth  of  the 
wliee!  IS  i  II.  1  he  w.iter  leaves  the  turtjiiic  with  a  velocity  of  6o  ft.  uer 
second,  heurmme  the  whirlin,;  viimitv  at  o.itist.  'he  rlnection  in 
which  the  water  leaves  the  turbine,  the  eliiciency,  and  the  liorse-power. 

Alls.  17.725  f/s;  7-   49';  ■'^'■;  214.S. 

13.  In  an  A.  F.  impulse  turbine  of  4  ft.  diameter,  1  It.  deep,  and  witli 
a  6-in.  width  (<f  opening  at  inlet  .iml  i.iitlet.the  etriciency  ('/)-•;  .8  ; 
/i  -  30  ,  )-  =  30  ;  l\  =  Ui.  Determine  the  inlet  lip  angle  (i(i,  the  elfec- 
tive  fall,  the  delivery  (0  (disregarding  thickness  of  vanesi,  the  11.1'.  .ind 
tlie  number  nf  levolutions  per  m  nuie. 

./«>.    .r  =  73°;    !.3'i6   It.;   27.39  cu.  't-  l»er   second;  O.322  11.1'.. 

14.  An  a.xi.ildlow  reaction  turbine  of  7  ft.  mean  diameter  passes  i^.S 
cu.  Ii.  lit  w.iter  per  second  under.!  loi.il  he, id  nf  1  3,3  It.,  the  ilepth  ot  the 
wheel  being  1  ft.  .\t  the  inlet-surfai  e  tlic  v.iiie-iip  is  vertical  and  the 
water  leaves  tlie  v  heel  vertically  If  the  inlet  W'dth  of  the  wheel  is  f  ft. 
,uid  the  ni.tiet  wclth  l^  It.,  tind  the  (Iirectmn  111  which  the  w.iter  enters 
tlie  wlicel.  tlie  direction  of  the  up  at  outlet,  the  iiiiet  and  outlet  areas, 
the  ill'.  Ill  the  turbine,  and  its  efhciency. 

.tns.  24'  4   ;   I';' 40  :   22  and  27J  sip  ft.;  285.535:  ..,4. 

15.  An  .i.\.,i!-llo'>v  turbine  is  to  be   used    lor  raising  water.      Expiaiii 


MICROCOPY    RESOLUTION    TEST    CHART 

ANSI  and  ISO  TEST  CHART  No    2' 


1.0 


I.I 


.25 


■  2f 

■  32 


1.4 


I  2.5 
[|||[  2.2 
[11  2.0 

[||  1.8 

1.6 


^      APPLIED  IIVMGE     Inc 


.Hi 


54» 


l.X.tMPLh.S. 


Iszj 


how  the  v,.„r.  should    l>c  a. ranged,   aiul    si.ow    how   tn  d<-t,T>n.,u.   the 

t'lh'  .     !lr  y. 

..■^.  In  an  A.  F.  iniijulse  turbine,  working  iiiulcr  a  iicad  of  loo  ft  tlie 
.i.rccljoM  „,  uh,ch  the  water  enters  at  the  mean  radius  makes  an  angle 
•1  =3  i6  with  the  vertical  and  leaves  the  wlieel  without  velocity  of 
wliirl.  The  .iepth  of  the  wheel  is  i  foot,  and  the  inlet  velocity  (v,)  is 
equal  to  the  linear  velocity  ,;/,)  of  tlie  wheeis  surface  at  the  mean 
■  ■.dius.  I  he  mean  diameter  of  the  wheel  is  3 J  ft.,  and  its  width  is  6  ins 
lind  the  hlade  angles  .it  inlet  and  outlet,  the  efiiciencv.  the  M.eeil  in 
revolutions  p.-r  minute,  the  amount  ,.f  w.iter  passing  through  the  turbme 
per  second,  and  the  1 1.1". 

.hu.  -,(,'  38';  64  52;  .7S;;  436,',  ,  404J  cu.  ft.:  3592J. 
17.  Water  is  delivered  to  an  ( ).  F.  turhine  at  a  radius  of  24  in.  with  i 
whirling  velocity  of  20  ft.  per  second,  and  leaves  in  a  levcrse  <lirecti.,n  at 
a  radius  ot  4  ft.  with  a  whirling  velocity  ol  10  ft.  per  second.  If  the  line.ir 
velocity  of  the  inlet-surface  is  20  ft.  per  second,  lind  the  he.id  eiiiiivalent 
to  the  work  done  in  drieing  the  wheel.  j,„   3^  ^  (^ 

IS.   An  r>iitwar<|.now  turbine  of  y.5  i„.  interii.,1  di.iniete'r  worksmider 
an  elbxtive  he.id  of  270  ft.     Find  the  speed  in  revolutions  per  minute 
assummg  tli.it  the  whirling  velocity  at  the  inlet-surface  relatively  to  the 
wheel  IS  ml  and  ili.it  the  efficiency  is  unitv.  Ans    '-4' 

19.  .\ii  oiiiward-llow  turbine,  whose  external  and  internal  di.iineters 
arc  6  It.  and  5J  ft.  respectively,  makes  2(,  revolutions  per  minute  umier 
an  etiective  hea.l  ,.f  4  ft.  Tne  water  enieis  the  wheel  in  a  .lirection 
making  an  angle  of  30  ly^  with  the  direction  of  motion  at  the  point  of 
entra.ce.  Deteimine  the  .iiigles  of  the  w.-j/W^r  -,;,„e  .a  ingros  and 
egress,  the  elTicfncy  being  .85.  Also  f.n.l  the  energy  per  pouixl  of 
w.iter  (  arned  away  by  the  water  as  it  leaves  the  turbme.  j.^ '  =  o. 

.        ,     ,  *'"'•   "  ^  '.'^'    2' ;  /J  =  2./  3S' ;   /,  h.Ahs. 

20.  A  ridial  .uiiw.,nl-(|,,w  turbine  ol  the  impulse  tv|.c  passes  8}  cu  ft 
of  wal.T  per  second  un.ler  an  elleciive  hea.l  of  5txj  It';  the  width  of  the 
wheel  IS  7i  in.  ;  ine  radius  t..  the  outlet-surlace  is  i.ij  limes  the  radius 
to  the  inlet-surface;  the  line.ir  velocity  of  the  inlet-surface  is  S;  ft  per 
•se.  ond  ;  the  direction  of  the  water  at  entrance  makes  an  angle  of  i7> 
with  the  wheel's  periphery.  Fin.l  ,„)  the  efTicicncy  ;  ,/-)  i|,e  lip  j,,,.,],.,  . 
U)tlie  areas  of  the  inlet- and  outlet-onlices.  neglecting  yfr../  hydraulic 
resistances,  and  jy.,y«,/ taking  these  resistances  int.,  a(  count  (,-^''-  =0) 

Ans.    First.   (,i)  .879,    ,/„  „=  14./   31'    and  ft  =  yy"  n\    'u) 
•  •535  «<]•   It.  and  .1291  stj.it.     Second.   (,/(  .767     ,/..,,  -  ,--''  ,,. 
and  fS  ■=  28^  55';  «•)  .176  sq.  ft.  and  .154  mj.  ft. 
.'I.  Construct  a  Fourncyron  turl>ine  for  a  fall  of  5  ft.  wui,  30  cu.  ft.  of 


w.iter 


I)er  sccontl.  ,r  =  80  .  ;.  =:  30'.  ~  = 

neglect  hyilraiiiie  resistances. 

A»s.  fi  =   lO'  42'  .   Ax  =  4.24  s<|.  ft. 
if  ^,  -  1.8  ft.,  then  </,  =-  ,/,    •     i.S  n 


\s. 


'\<\\ 


r,. 


W,    -   5.8189  ft.;  >/ —  .915; 


HXAMPLES. 

In  an  impulse  ,iutwar.i-(i(,w  turbine  cf  lo  B.II.P 


a    head   nl  9   tt.,  y  =z 


</i   =  .2>-| 


i5o"  —  <j  = 


'Ih 


37i";  li   =45';  9( 


543 


.woiking  under 


ftUrance.      Finil    tli 


ere  is  a  loss  of  5  per  cenl  due  to  frict 


>» 


r,)  =t, 


cffic. 


Uk 


i'lri  in  the  velocitv  .it 


secoiul,  and  the  diameter  of  the  turij 


if 


water    [i.i^sed    per 


23.  A   turl)inc   dehvers 


I  vsr 


'9  cu.  ft. 


I    eu.   ft.  of   water    per   second.      Th 


caves  the  outlet  peripliery  radially  {vj'  =r  01.     Tl 
idial  (<r  =  9<o-j,     The  direction  of  infl 


49  ft. 
B   'vater 


wiieel's  periphery.     T'le  radius  of 
revohitions  per  tiiiiiuie  is  100.      If 


le  vane-lip  at  inlet  is 
)\v  makes  an  angle  of  60"  with  tl 


Lead  and  the  ellcetive  work 


inlet  surface  is   2  ft.     Tlie  number  of 
the  rineieiicy  is  90  per  cent,  lind   the 


24.  One  cubic  foot 


.'Ins.    15.243  ft.;   1.5625  H.P 


V  heel  of  2  ft.  cxter 


w.iter  per  second  enters  a  radial  ().[■" 


with  the  radius,  and  leav'e 


nal  and    il    it.  internal    diameter 


i|Hilse 


It   an   angle    of  60 


riie  pcrlplieral  speed  at  tiie  outlet-surf 


s  without  whirl.     The  effective  head  is  400 


terminc    <x.v,.  the  outlet  and    inlet 


ice  is  20  4  3  ft.  per  see 


eiric 


lencv. 


A  US. 


5.  In  a  radial-flow  le.ict 


15  sq.  Ins.,  i.S  sq.  ins.;   183  ir 


areas   and    depths,    the  11. 1',    aiul 


.39  ins.;  12,8;  .28. 


i<jn  lurbi'ie  with  radial  inlet-lips,  if  ,/,  =  j,/, 


and  ;-  :=  tan-  '  4,  show  that  the  recq)ro<  al  of  the  etricie 
if  the  whirling  velocitv  at  outlet  is  nil 


1« 
ncy  is  I  4-  tan  />' 


26.  An  < ).  F.  ir 


eter  pas.ses  100  cu.  ft.  of  water 


ipulse  turbine  of  3^  ft.  cxteri 


;ft. 


interior  diani- 


eiitranct 


(>er  second  under  a  head  of  62;  ft. 


the  direction  of   the  water   makes  an    angle  of   30    witl 


»i 


p-'riphery.     If   the  relaliv 
•  letermine  the  diieclion  and 


leaving  the  wheel,  the  effi 

ute.     Disregard  hydraulic  resistan_ 

.  /'/(    If   </,  =  ,/,  ,  r,  =  91.065  ft.  per  sec. ;  1?  = 


find    peripheral    speeds  at  outlet 
lagnirude  of  the  velocitv  of  i| 


aie  equal, 
le  water  on 


iciency.  a,,d  the  speed  in  revolutions  per 


min- 


A 


. .  J'.  =  iog.435 


•0°  14}': '/  =  79;A'=  734.8. 


;  6  =  66°  02' ;  n  = 


r.idial   impul.se   turbine  passes  KJ  cu.  ft.   of 


.70:  A- =734.8. 


ellective  head  (jf  5(10  ft.     The  d 
at  ij°  to  the  wheel's  periphery. 
■■■7  ft.  per  second 


water    under   an 
reciion  of  the  entering  water  is  inclined 
The  linear  speed  <if  the  inlet-surface  is 


A.ssuming  that  the  velocitv  of  wliiil  at 


iind  disreg.uding  hvdraulic  resistances,  find  („)  tl 


velocity  with    which  the  water  enters  the  wheel 


the  outlet  is 


lie  water  as  it  leaves  the  wheel  ;  (,/(  th 
ing  and  outflowing  stream;    (/•)  the  di 
•  /)  the  power  of  the  turbine. 

The  ladii  of  the   inlet-  and  out 
spectively.     Find  {f^'\  the  direction  of  th 


e  efluiency;  {b)  the 
(' )   the    velocity    of 


e  sectional  areas  of  the  infl. 


the  v.iTie- 


ip  at   mil  I 


et -surfaces  arc  4I}  ft.  and  4Ji  ft. 


le  vane  edge  at  outlet. 


Ahs.  (</)  .879,  (I,)  189.31  ft.  per  sec. ;  (,)  65 


Uit 


86  ft. 


\^y',(>  sq.  ft..  .129  sq.  ft. ;  (f)  a  =  ur/  3 


[M-r  sec, 

r 


28.  1 1 


(yU75  43  H.l'.;  (v) /»  -  41'' 3'. 
preceding  example  show  how  the    results  are  afTected  by 


ffl 

ff) 

';  v.T; 

^ 


fc4 


544 


FW-IMPLl^S. 


and    '    of  il:e  (Iciiih  'if  tin-  outiot- 
3^ 


;H-' 


t.ikiiig  .<;4  a^  llie  cueiricuMn  of  velocity  in  c.iU  ui  .uni;  the  vt^lonty  with 

which    the  u.itcT  t-iucrs    ihr    ulicrl,  ami    .issuniiMi;   that   ^^^     ^        i-    tlio 

(riciional  loss  ..f  heaii  in  the  passages. 

.l„s.  i,M  .826;  (/>)  177-955  fi-  per  sec;  (» )  3''-734''^  f'-  I'C  sec; 
(,/)  .103  sq.  ft.;  .zy-A-  sq.  ft.  ;  («■)  "■  =  14;'  57'; 
(/)  440.9  11. P.  ;  (.VI  fi  =  0°  54'- 
2y.    Ill  an  I.l'.  turbine  the  radius  of  the  inlet-surface  is  twice  that  of 
the  outlet-surface;  the  linear  velocity   of  the  inlil-siirface   is  one   half 
that  due  to  the  head  ;  the  water  enters  the  wheel  witii  a  velocity  of  flow 
(7//)  equal  to  one  eighth  that  due  to  the  head,  and  the  sectional  area  of 
the  water-way  is  constant  fioin  inlet  to  outlet.     Kind  the  angle  between 
the  discharging  lip  of  the  vane  and  the  wheel's  periphery,  the  whirling 
velocitv  at  the  outlet-surf.ice  being  nil.  •''"•  <-ot     '  J. 

30.    In  a  vorte.x  turbine  the  depth  of  the   inlet-nrnices  is  one  c  igntli 

of  the  diameter  of  the  wheel  |^     ^^'1  ' 

orilices.     Thewidtliofthewlirulisonetenth.il   the  (ti.uneter   ^=    ^^j. 

The  inlet-lip  of  the  vanes  is  radial,  and  the  water  enters  at  an  , ingle  of 
30°  with  the  inlet  peripherv.  Find  tlie  size,  sp.'ed,  and  ellieieney  of  the 
turbine  in  terms  of  the  supply  ..f  w.iter  (J  and  the  etiective  head  //. 
Also  find  the  direction  of  the  outlet  edge  of  the  v.ims. 

An-.    1.    .Assume  I'-..,"  =  o.     Then  r,  ~  .45K      ; 

//' 

//» 

No.  of  revolutunis  per  ininute  ~  l"9-5     .  ; 

<2 
n  =  .863;   fi  =  35"  II'. 

<>" 
II.    Assume  u,  =   r,.      Ihen  r,  =  •44-^,; 

No.  of  revolutions   per    niiiuite  -   122.30     ,; 

I  r 

>!  -L  .8146;    fi  ---  44'  4^'. 

31.  .\  vortex  turbine,  with  a  wheel  of  2  ft.  di.imeterand  6  ins.  breadth, 
passes  10  cu.  ft.  of  water  per  second  under  a  lie.ul  i>f  32  feet.  Find  the 
inclination  of  the  guides  and  the  power  of  the  turbine.  Assume 
Ml  -  W  ,  <»  =  fA>*,  and  the  efficiency  =  1.  Ans.  5°  41' ;  ./>i'i  H.P. 

32.  .\n  inward-How  turbine  has  an  internal  r.idius  of  12  ins.  and  an 
external  radius  of  24  ins.  ;  the  water  enters  at  15*  with  the  tangent  to 
the  circumference,  and  is  dischnrgeil  railiallv;  the  velocity  of  radial  flow 
is  5  ft.  at  both  cirrumferenres;  the  velix  ily  of  outer  periphery  of  wheel 
is  16  ft.  per  second.  Fiml  the  angles  of  the  vanes  at  the  inner  and 
outer  circumferences,  and  the  usrfnl  work  done  per  pound  of  fluid. 

Am.  /<  ^  32';  r/=  118°  1';  9.35  ft. -lbs. 


f-:\.-iMPij:s. 


545 


33.  For  a  supply  of  64  <ii.  ft.  \>  t  second,  under  a  liead  of  Si  ft., 
deleriniiie  the  speed,  .size,  H  1'.,  aU'l  otiu  irncy  of  a' vortex  tiirl)iiie  in 
•.vliicli  if,  ~-  r,  =  yi,  —  5  x  width  of  wiicel,  as<iimiiii;  tliat  tlnTr  i>  no 
vcloi:i:y  of  whirl  at  outlet. 

34.  A  i.idial  I.  !•'.  re.ii'tlon  turbine,  v.itli  ■ 'r  \v:tlio\u  dr.ift-pipc.  p.isses 
113  cu.  ft.  of  waier  under  an  ctTeclive  liead  of  1  ^  ft.  The  radius  of  the 
iidet-surface  is  1 .1^19  times  the  raiiius  of  the  ouil'-t-'-urf.ice,  and  t  .■■  L.tio 
of  L  le  outlet  to  the  iiilcl  area  is  .92.  The  vane-lip  at  outlet  makes 
an  aii!.;le  of  I  5"  with  the  wheel's  periphi;ry,  and  the  water  enters  at  an 
.inL;lc  of  I :'°  with  the  wheel's  periphery.  The  sectional  arc.i  of  the 
<lraft-tube  (if  there  isone)atthe  point  of  disciiirp;e  is  1.035  times  the  mi - 
tional  area  of  the  outlet-oritice.  Show  ih;,t  tlie  useful  work  per  pound  of 
water  is  1  I.I  17  ft. -lbs.,  and  that  the  work  consumed  in  hydraulic  re.,isiance 
i.\ri.  !■'.,  pai;e  531)  is  nearly  r.SSi  ft. -lbs,  ;  .ilso  find  ./i,  . /j,  7'j,  and  the 
eiricieiu  \. 

Alls,  ill)  :S.J975  S().  ft.;  26.03  sij,  ft.;  (/'I  4.  U  "•  I""''  sec,  ..S55. 
^iv  In  the  prrcedui^  example,  if  the  r;idius  of  the  oui  let-surt.ice  is  4 
It.,  rind  ill)  the  s|n-ed  of  the  wheel  in  revolutions  per  ininuie  ,  also  find 
\i>)  the-  depth  of  the  wheel  at  inlet  and  outlet,  the  guide-vanes  beini;  40 
and  the  wheel-vanes  41  in  number,  and  the  thickness  of  the  former  beiiifj 
,^,,  inch  ,111(1  of  the  latter  |  in<h.         .-///.v.  ',1)  3.S.65ft;  (/O  1.23  ft.,    1. 3;  ft. 

36.  In  exaiiiple  34  tliid   the  eflicii'iicy  if  the  di, muter  of  tin'  dr.ift 
tube  is  made  the  same  as  the  di.imetcr  of  the  oiitlet-siirf.ice,  the   lower 
cdj^e  of  the  tube  beiii).;  rounded.     W'li.it  will   be  tin:  "  loss  in  shock  "  in 
the  tube  per  (xiund  of  water-'  .Ins.  .864;  .077  ft.-!bs. 

37.  .Xn  iiiw.ird-llow  turliiiie  has  .ill  extiTual  diameter  of  3  ft.  .iiid  .111 
internal  diaineti  r  of  2  ft.  It  passes  I2cu.  ft.  of  water  per  second  under 
an  ellective  head  of  40  ft.  The  water  enters  the  wheel  at  an  anyle  of  30' 
with  the  wheel's  periphery,  and  the  depth  of  the  oiitlel-orilices  is  twice 
the  depth  of  the  iiilet-ordices.  The  etlK  iency  of  the  turbine  is  .9.  I)is- 
resardiiifj  friciion,  lind  (<i)  tlie  vane-angles  at  inlet  and  outlet ;  (/5)  the 
velocity  with  which  the  water  leaves  the  wheel ;  (< ,  the  speed  of  the  tur- 
tiine  ill  revolutions  (kt  minute;  o/)  the  velocity  with  which  the  water 
Cillers  the  vvhi'el  ;  ((■)  the  areas  of  t  he  outlet- and  iidet-orili(  c^  ;  i/)the 
power  of  the  tuibine  (7,,."  =  o). 

Alls.  {,!)  a  =  105- 09',  li  —  35"  3;'  ;  (  h\  lb  ft.  per  sec;  (< )  198.39; 
(,/(  42I  ft.  per  sec;  to  .5(125  sq.  (t.  ;  .75  sq.  ft. ;  (/i  49,',  III'. 
3.S.  In  an  inward-flow  re. ictioii  turbine  of  6.27  H.I',  the  radial  \eio(- 
ily  of  flow  is  constant  from  inlet  to  outlet  and  is  12  ft.  per  second.  1  he 
water,  with  a  velocity  of  fiofl.  p»T second, enters  at  1 1'  32'  with  the  wheel's 
periphery,  which  h.is  a  linear  speed  of  50  ft.  per  secoiwl.  The  diameters 
of  the  outlet- .ind  inlel-surf.ices  are  1  and  2  ft.  respectively  I'ind  the  lip 
angles,  the  head,  the ciriciency, and  the  rjuantiiy  of  watei  passing  through 
the  turbine  per  second. 

Ans.   It  -  126    12';  /J  =  151'  19' ;  91.S6  ft.;  6o?  ,   1  c  u.  I'.. 


r.n  i 


!;■     ■ 


? 


te 


fe:4 


■;4') 


HX/I.Mrl.FS. 


\g.  An  iiiwar.l-fl'.w  radial  nni,uisc  tuibine  of  4-5  ft.  and  4  *'■  cxternat 
ai.d  nu<  iiKil  i.i.lii  parses  H\  (  u.  II.  of  water  per  SfOJiul  uiicU'i  an  ,  llcctive 
luad  of  5')d  It.  riie  ilwe.tr. n  ol  ihc  en'.enn-  w.iu-r  is  m.  lined  .a  17°  l'- 
ti.e  wheels  penpherr  .  and  the  wheel  lia^  tlie  >anie  d.^ilh  at  the  mlel- 
aiul  outlet-surfaces.  '  ll  li.e  per.pheral  speed  at  tlie  ,  nlU  t->urlace  («.) 
is  equal  to  the  rel.itivc  veioeitv  of  the  water  (  / '.)  witli  n-pect  to  the 
wheel,  fmrl  (,;)  the  efficiency  ;  (M  the  speed  of  the  turhin.-  m  revolutions 
p.r  minute;  (,  1  the  sectional  areas  ul  the  stream  at  mlet  and  outlet  : 
(,/)lhe  dnection  i.f  the  vanc-outl-t  e,!-.-;  i,,  the  velocity  of  the  water 
as  it  leaves  the  w  heel 


/■)  the  power  o|  the  turhi 


It 


ft.. 


,3  11  1'. 
lesulls    w  ill     1  ( 


.•;«.«.  i.i)  .873;  (/')  20(;.94;  10  .Ij.',;" 
(,/l  /<  =  43  2';  (,■)  67.39  ft.  |.er  secon<l  ;  1/  14:-. 
40.  In  the  preceduif;  c.\aiuple  examine  how  the 
alTected  when  hv<!raulic  resistances  are  tiken  into  account,  allowing  .94 
as  a  coelticient  Iif  velocitv  for  the  w.itrr  on  entering  the  wheel,  and  a- 
snimn-  that  the  hea.l  cipiivalent  to  the  relative  velocity  (  f.i  on  le.ivicu 
the  wheel  i<  increased  hy  10  per  cent. 

./;/v.  i,i).S(,5;  {/')  193. 1'S;  revolutions  per  minute  ;  (()  .163  s(|   t'  . 
.14;  MP  ft.:(./i>^4('    18';  (<■)  63.053  ft.  per  second  :  (/)  4f'7-''^  1  ' '  i' 
41      .\ii    I.    1-.    turbine  of  4  ft.    extern. d    diameter    works    under  . 
clfecliw    luMd  of  230  ft.      Find  the  speed  of  the  wheel  in  revolutions  p. 
minute,  ;■  '  being  o,  the  etlkiency  //;;//r.  and  d  -  '«>=.  yhis.  4-7- 

42.  An  I.  F.  turbine  of  4  ft.  external  and  3  ft.  inlernal  .li.imit.  1 
makes  3(10  revolution^  p.  r  minute.  The  sectional  area  of  flow  is  3  sq  I!. 
and  is  the  same  in  everv  part  of  the  turbine.  The  direction  of  the  in- 
flowing water  m.ikes  an  an-le  of  30    with  the  wheel  s  periphery.    Assum- 

il  the  oiitlel-suilai  e   1-.   nil,  lind  ui)  tlf 
the  deliveiv  in  cidiu    P  ei    per  minute. 


iny  that   tlie   whirhni;  velocity 
eiri<iency;  (/')  the  11. P.  ;  and  (c 
The  total  held  is  200  ft. 


.his.  (.1)  .SO  ;  (/''  :476.«  :  (ci  7593- 


43.   An  inward-flow  turbine  beinj,'  required  for  an  available  head   oi 
20  ft.  and   a  dischai^e  of  800  cu.  ft.  per  minute,  determine  (.1)  the  si/c 


iiid  (/'}  the  si)eed  of  the  whee 
ivliiel 

depth 


1,1  the  mtlin.itions  of  the  guide-  and 
ibini', 


wheelvanes  ;  and  (./)  the  efficieiKy  ol  the  tin  bine,  „ssuming  ; 
,1   wheel  ;  ■;•  --   ^24'//;  JN.."  =  o.  .t  =  90'.  and  </,  ==  -''. 


Afis.  {.!)  r,  =  .487  ft..  »  ,  -.  .974  't-:  "'"  -4"  revolutions  per   m 
ulc ;  (i )  y  ^  xo'  it'./i  =  }(<  8  ;  {J,  93J  \^r  cent. 
44.  A  vortex  turbine  passes  O  cu.  ft.  of  water  per  second   under 


.neciive  head  of  //  ft.    The  inlet-lip  of  the  vanes  is  radial,  and  the  din  ( 

nteriiii;   uatei    iiLike-   m   aiiuh-  of    2o*  17'  with   the  wheel- 

t/A'         ,    n/). 
he  are.isol  the  mlel- ,it;d  ouMct-orifices  are       ^^      ami 

ih  oj  the  wheel  is   -'.  /',  beiiii;  the  flianu'tcr  ol 


tloll    ol    lIU' 

per'pliei  \ 


i< 


spe(  lively,  and  llu   w  u' 


the  inlet-surface.     If   tlie  whirling  velocitv   at  the  ontlet-siiiface    is  nil. 
find  1.(1  the  efliciency  ;  (/•)  the  (liicclujii  ot  llic  outlet  edge  ol  the  vane  ; 


z.".v./,v;/v./..s. 


547 


(t)  the  velocitv  with  which  the  wau-r  enters  and  leaves  the  wheel;  (,/'i 
the  speed  of  the  wheel  in  revDlutions  per  niinute;  k'i  the  (ii.iincteis  nf 
the  iniet-  .md  >  .utiet-surl.u  es. 

.■hii.  iii)  .>'by.{/'\  //'  —  35"  lo':  u)  ii.ndy;//'.   i.tf'i-jJl* , 

■» 


{it I    Iov,5J        ;    I,  I    .lyi;  -    . 
'J'  It- 


'll 


'2 
11 


45.  A  Viirlex  tiiibine  i,.i,,-.Ls  11  ,  u.  It,  uf  water  per  second  umier  ,1 
he.ul  ol  35  It.  .  the  diameter  ol  liic  oullet-surtace  is  1  It.  and  its  breadtii 
6  ins.  Find  tlie  power  of  the  turbine,  di.sreyarding  friction  and  asMiininj; 
that  ilie  uiiirlin-  velocity  at  the  oiitiet-siirfacc  is  ml. 

Aiu.  4V  5  1 1. 1'. 

4I'.  Kind  llie  11.  P.  developed  by  an  1.  I',  t  ui  t)iMe.  of  3  it.  .-.xternal 
and  \\  ft.  intcrnai  diameter,  pa-siny  .>)o  tons  of  water  perhoui.  The 
velocity  of  whirl  at  inlet  (f,.')  is  equal  to  that  of  the  periphery  and  i^  45 
It.  |)er  second:  the  outlet  veloritv  of  whirl  is  lO?  ft.  per  seconl. 

.Uu.  4O4. 

47.  A  turbine  with  radial  vanes  i)asses  3600  i;ailons  per  liour  under 
an  effective  liead  of  36  ft.  Find  the  peripheral  speed  and  tlie  inlet  ana 
SI)  th.a  tile  elfkiency  may  be  a  nia.\imuni. 


1 

«• 

M\ 

■ 

hB  ''     ti  <t   ' 

1 

Hi 

1 

it 

^  s 

il! 

i'l 

^  i. 

1  j 

I  il 

i 

fi  »; 

!    i 

i!!: 


0 


^^ 


Ua 


CIIAPll.R    \'i;i. 

(•K\TK!IT(,  \l.    rt'Ml'S. 

I.  General  Statement.  If  an  livdr.uiliL  nmtnr  is  ilrivc-n  in 
thr  r.'vrr-c  ilirccti'iii,  .iml  -^upiiHcd  with  water  at  th.-  point  iVom 
\Khkh  the  w.itcr  ..rr^inallv  iirncccdc'd.  llu-  ni.'t..r  hocnmcs  ri 
pump.  All  turliinrs  arc  rcvcr-iliU-.  and  nia\-  tluTcforc  he 
vonvcrtcd  into  punip<,  tint  no  jjinnp  lias  yet  hern  constriR  ted 
»)f  an  inward-How  t\pc.  IIk-  ordi:iar\-  ccntrit'uL^al  luimp  is  an 
«iut\\ard,-llo\\   mat. diinc. 

15.-tori-  t!i>'  pump  e.m  In-  I'ut  into  action  it  mu-t  he  llllcd, 
and  this  can  he  et'tected  throu-h  an  opcnini,'  (closed  hy  a  phi;^) 
in  the  casini,^  when  tlie  puiiip  is  under  water,  or.  il  tiie  pump 
i-,  .d)ove  \vater,  h\-  ue.itin-  ,i  v.icuuni  in  the  punip-c.ise  hy 
means  of  .u\  ,iir-pump  or  ,i  -team-i.t  i)nnp'.  v\hen  the  w.iter 
must  necessaril)-  rise  in  tlie  suctioi'-tuhe. 

At  first  the  water  rotates  ,as  ,i  s,,li,l  mass,  .md  de]i\er>- 
tonnnenc-  when  the    -.jjeed  is  such    that  the    head  due  to  i  en- 

trifu-.d  force    '"■   7,"'^   exceeds  tlie  lift.      Tin.  speed  nia\-  he 
aftern.irds  reduced,  jirovidin-  .i  i)ortion  of  the  eneri;\-  is  utilized 

;it  e\it. 

As  soon  ,is  the  pumi>.  wliich  is  keyed  on  to  a  sh;'ft  driven 
by  ;i  belt  or  oilier  lUi-an^.  o.ninuiKes  t  >  \v..rK.  the  water  rises 
in  the  suctioii-tube  ami  enters  the  eye  of  the  pump-disc  on  one 
side,  or  divide-,  .md  eiiteis  on  both  sides. 

As  in  turbines,  tlic  wlieel-bi.ide  tips  are  so  curved  as  to 
receive,  at  a  -.pecitiod  normal  speed.  t!ie  intlowini,'  water  with- 
out   shock.      The    water   leaves  tlie   disc    with   a   more  or  less 

548 


Cr.NTKIhl  (,.-11    II  \ll\s. 


549 


Con:-i(li.T;iblc  \  Lloi.ity,  aini  iinpiii;^^-  iipdii  t'lic  tluiil  nia'^s  llnuini; 
arDuml  t!u-  \  nlutf,  (ir  spiral  ta>iiiL;  ^iirnnindiiiL;  tin-  (ti--v  ,  ti  iw  ards 
tlic    discharL;i-pipr.         I'liis    volutL     ^hniild     ha\(.     a     section 


Fic.  313- 


j^raiiiuilly  iiicrtasini^  to  tlic  point  nf  discliarLjc,  in  order  tliat 
tlif  dcli\cr\'  across  any  transverse  section  of  tlic  \oliite  ma\' bo 
iinitorni  I  his  \iiliite  is  als'  so  dcsi^notl  as  to  cuniiKl  rotation 
in  one  direction  <>nl\,  witn  a  M'lncit)-  iDne-^pondiiiL;  id  tin- 
vclocitv  i>t  \\  liiri  (:  ,     1  <in  leaxini;  tli<   Ian.       Tin  ve  an- examples 


4 

M 


5 


mt 


5;o  (:i-srRiir(,.-a  ri  mi^s. 

'.f  pump-  in  which  the  ckhvci)-  i->  i  i'lLAtcd  in  all  dircctinn-,  and 
tlio  water  i-  -uided  In  the  (nitht  h\  a  mmibcr  >  it'  -piral  lil.idcs. 
A  ci'iitnt'u;^al  pump  i--  more  ccntioinical  and  K;ss  c<wtly 
for  -hort  lilt-  than  a  iri.i])rc  icatin-  i)unip.  and.  has  iK-.-n  known 
t,,  L;i\<'  L;i")d  and  i-idnnniii.   n  -nit-  tur  lifts  a-  Ljfcat  as  40  tt. 

With  i- impnund  n  innt"u-al    pinnps  \i  r_v  miK  h    -1.   iter  lilts 
are  eci)ni>mie.dl\'  pussibK-. 

'I'herc  .u-e  three  main  dilTereiucs  lietw  een  1 1  ntnUi'_;al  jninip- 
and  turljine-  : 

l-t  The  -rosv  lilt  with  a  pninpi-.  -reater.  •  .w  aeeount  of 
tVietional  rcsi-t.uue-,  than  the  tall  m  the  ea^-  '>t  a  turbine. 

2i\.  Tile  water  Cillers  the  punn'-lan  Jianibei  wiiinu't  an\- 
\(d(ieit\'  "I  w  hii'l  r  .'  -  O  .  ,ind.  K  a\  e-  the  tan  with  a  \  ^lueit)-  <  •( 
\\  liiil  I  ,-■  1  u  InJi  shddd  he  red  need  X>i  a  mimniuin  ;:\  tin-  act  (>( 
liftin-.  hut  whieii  i-  In-  iM  mean-  <mall.  In  a  tMrhnr.  ,  ..n  the 
other  hand,  tiie  water  ha-  a  eMii-ider.djle  v.-j,  „:ity  ot"  w  hirl  1,-^,') 
at  entrance,  while  at  e\itthe  Ml^eit)  .it'whirl  r  .,  i- redueed 
t<>  a  mininnnn.  and  is  -enei-all\-  ////. 

",d.  In  a  turhnie  t!ie  dir<  etion  ol'  tin  >\at''r  a-  it  ll.w-  into 
tlu:  wllecl  i-  CMHtiolled  1)\-  -  nide-hlade,  ,  w  herea-  m  the  e.i-c 
ot"  a  pump,  the  dir>ction  of  the  water.  .1-  u  llow  -  tiwards  the 
<lischar.L;e-pipe.  i-  controlled  1)\  ,1  -inj^le  ;^uided)l  id'-.  whiJi 
t'orm-  the  outer  surface  of"  the  \ointe.  oi'  i  handier,  int  ■  v.lnch 
the  w.iter  t1ow-  on  leaxiuL;  th<'  fan. 


in..  314.      Kxpi-ritiiciil.il   Centrifug.il    I'ump   in    itic  llytr.iuiir  I.,itiiiratorjr, 

M'  ('.ill  I'liivcisav. 

l-.\]K-rnnent  -e^in-  to  mdiciti   tiiat  tin   el'tu  leiu  >•  of  ,1  centrif- 
US^.al  pump  inere,i>cs  a-  tin    inh  t-tip  ain^h   dnnnn-lu-,   .iin!  tltat 


i^raUUil.] 


-.rsTRiH  r..n  ri'MPS. 


it   i-,   iheict'orc  .11 


l\;uUa.LJ,c 


HI-    I"    111. I 


kc  llii-^   an- 


DD' 


lo   a-   -niali  .1-  1- 


]i!ai.tKal)lc 


1,1c,   buV    wpiiiKii 


iv   ..n  till--  piMiil  I. 


lilTcr.      'n"-'  '^'''l   '"'"'"' 


,,,,  it  ,.  doul.tnil  uhctlK.  th..  onhiiaiy  iui-tluM,  ,,1     a^i^^^ 

::  =  ,o'iat,n..t.itli.a,t.lio.U  i.cv.ii  a,,,n..nnatclyc,.,..^ 

'       TlcuUct^.l.cayan.itlu.M.,-caK..U,c,>u,np.cni.uK 
,vhc  in.rca.c.l  hv  the  u..   ..,  a  suai,.n-tutu.  .itli  a  „a.lual,> 
\       '".,.,,;,,■,.-,.  a  tube    intl,     fnnii  ...■   tlu.  ini^tuni  ... 


(liniiiii'^lii'if;  --eel  11)11.  '■.- 


s  1),    ..litaiiiiil  In-  i;ivini; 


a  cone.      A  ^tilUre.itcr.ulx.uit.iL;.    i".i> 
t,K-a,.cl,a„.-n.,na,r.uluallyincn-aMn,.ect,nn       In  tlii^  .  .v>e 
'.    ,e,t;,.r.l!^cliai-,e,i-a.h,ally.li,nini^lu.an.,  ■',.    i-^-'n. 

,      i.   ,;.p,,i-t,,.natelvin.rea.ca,    ...   th..,    then    .   a,ain.. 

av'aihll.le     ,.,■     nure,.,,..      the     ,.„„,„„     , ei  1  he 

t,eitvinthe.i.char.e-pi,.c.h,,uU,    ii..thet.,,,,voa^ 

■Ul      1,,.-,  of  eliei    '\-  denelMlh    s pe.lk- 
,r-i\-  le.ul  t.i  .1  vere  -^en-ioK'  ,<.->  "i  mLi.,). 

-             ,      ;        f\  ,,r,  tt     ,.er    .....n.lh.i.  been    1.  ...niu  t..  -ive 
ini;.  .1  veli>eit\   ..t   ^^  t..  c  n.  ]". 

ilir   ni.i.^t  ta\  .  M'.lbh'   1  e--lKt-. 

''"  bv.oni.-   .e..h.,nt,e.   th.it    an    a>Ua,it,i,e    may 


It   i-.  ehniiK 


,; ....:/....■, 1-'.-    ,n-"..'.- >■;;;;;-■; 

,,p.-,.,l    ih.it    the    uater    .h^.h.iu-  '\   h.-m    t.u 

-U.inth,.Ui,>inbei.an.,that.M...-''n-;N.b.,ipice^ 
.   thu--...nve:,e.liiU.,i.re.M,nene,-,y.  which  u.,uKl..th.nMe 

:.,„.., V    ...tea    in    e.hhe.   m    th-    ^..h,te   ..i"   a.chai-.c-l.pe 

LXu\.n..,-^  The.. Uei-  leave.  tl.  —  ex-eh,iinber 

1  m,    a    a,,nin.he.,    .h.ilni,    veh,e.t>  ;Une,,    e.unj,.t       e    v^^^^ 

diUeunt    in.hre.t„.n    ,uul  nia.nitiule   Ironi    the    vch.ut^    ■      th. 

....i.ateMnthev„b.t.,        Ih..  v,  .„ex-eha,nbe,- .>  ..  .niet.i    e. 

.     ,        .,.    ...:.,..  1.1  „■„..., ,n,,wniv  the.hi-eai..n..thvev..itex 


Pi' 


a-    In    ine\ciU    ine^uhir 
ait'feient    ronditiuiis    from 


,,,,vule.l  with. ;.iule-l>heleO,.n.. will- 
.tre,im-lme-  . e.iuMH-ail.n     -l'"'^!- 

iii.ili..ii. 

Ceiitrihi'^.il  i>unv  uoik  uiuler 
tni-bin.-.,  ami  hence  thei-e.ue  .  ,„Te.,.ondi,u;  .hlfevences  nc.-cs- 
Xm  their  dcsi.n,  ,  hey  .  o.U  host  ,0,-  the  paiti.aHi.to, 
Xh  they  are  aes.,nc.l.  .mcl  .my  v.u-,ati..n  ii-om  thi- hU  ..U.CS 

arapul  ie(lncti.:n  m  the  elticiency. 


ft! 


m 


fc::? 


55^ 


w.v-//ys/\  o/   cESTRihuc.-i!.  riwir. 
y in.  U5. 

.-       UPPER   WtTER   LE^EL   —  ■ 


■"  --   LOWER   AA-ER  -E/EL ^ 


Vtn.  316. 


^\.1IYSIS  OF  CHNTRIH  (j.4l.   t'lMP. 


55i 


2.  Analysis    of    the   Centrifugal    Pump I)csi<jiiatc  the 

velocities,  iuii;les,  ;uui    i)ri,>'-urc-hca('i>  ,it  thi-  inlet-  ami  nntlet- 
SLirfaccs  c it  till    wheel,  l-'ii,'^^    ^15  ami   ^\<>.  \>y  the  same  sj-mbols 

a^  in  the  i.i'-e  ciftll"  tiii'hiiir.    .Art.  4. 

Let  0  !»■  tin-  i!eli\i.i\-  of  the  ininip  in  cii.  ft.  \Kf  sec. 

I.i-l  //^  he  the  .L;r(iss  hit  iiKhiilinL;  tlie  head  eijiiix  ;ilent  to  thi- 
total    h_\'clraulic    resistance   i//,\.    the    actual    hit    ,//,,),    and   the 

head    i-iHii\,dent    tn    the     xchicite     1  if   iIcli\\',-\-    ir',),     vi/.,       '   . 

Th.  n 

//.  -  ^.  +  fl.  +  ^. 

//,     include--     the     head.s     ei]iii\  alnit     tn     tlu-     ii:->ist,ince     in     the 
siicti(in-lilj(e  i//,    .   in  tin;    deli\ery-])i]>i       //,i,   and    in   tin-    uhce! 
])assages  •  h ,\,  sd  that 

''r^l',-\-l'A-l'y 


Fig.  317. 


i-. 


Fig.  318. 


//^  includes  the  heii^ht  I'f  snctinn  1//  i.md  thr  hei^dit  m!' d.eli\  1. 1; 
{hj),  so  that 


.-ii« 


p 


f^ 


554 


lAlssl.S   IS  HYIiK.tUI'     kl^lSl.-lNi  h 

,,„.  luiMu  nf  sr-tion  :/,^  U  ^cm.ndly  1.1.  n  t-.  U-  the 
vcrt,.-aia,..^,K.  iK-tuo.ntiK.  lnv,.M  uatc-Kv.l  and  ll,cax,. 
of  tlH-  l.nnp.  la.t  tlu.  ,-.  nu,.mxt  an,l  may  l-ul  to  snaou. 
,,,„,.  -11,;  tnu.hu^lu  .,t-^uau,n,  i.ctlu.  hci,rlUt.,whKh 
,,,,,,,,,,  „„M  In  rai^.cl  l.lon  th.  lunup  u  ,11  o.mnuMK.  f . 
,1,,  w.uk,  -h..uM  In.  ,„.a^>uv.l  fn.n  th.  loucr  uat.r-Kwl  to 
UK.  top  oftlu-  nui-cUo,-  ov  to  .h.  to,,  of  thr  uh,.l  a.vor.ini:.  as 

t!u'  asi-.fth.    lunnpi.  !,„ri/onta!  or  v.rtKal. 

Tin-   aau.l  lo..  ,n   by.lraulK   n^i^tana-.   iKt.-.n    th-    mu- 
t,on-Uv>.l    an.l    tin    ...    of   tin    pun;,   .nay   h.   .Ktcnnnu.     by 

tin  f.llovMH-  I'Ktlioil  su-;.v-t.-.l  hy 
Allu-rt  1  Hall.  A  Ion-  -au-i-- 
-1,1--  .1/.'.  \^'tli  a  wi-t-iron  .ap  I'K 
r^  filtcl  into  tlu  t..p  ol  tlir  s.ution- 
,„,„  Ih,    XV. Iter   ri-^c.    in    tli.-   tul.o 

t,,  a  ^.vtam  l.x<l  ■'''■  •""'  ^''''  l"*-""- 
sun-  ni  tin-  tnl-r  .an  !»  .In.'alv 
nu-a-^uiad  1)>  niran-  of  tho  -an-r  >,. 
If///,,  is  tlio  h.Hdin.tru  luMil  .nn! 
//  tin-  -iiniM'-i.  .ulnn.;.  ni  t<<'t,  tlfn 
//  -  //  1-  the  .u  tn.il  lU  nanni  In  ,ul 
at  ,1,!.  II.  nc.  if  //  1-  ''!'■  -l''">- 
Iu.hI.  I.'-.,  til.  \.itu,il  .iwt.uK.-  Ih'- 
tA>-cn  til.'  Mut>.,n-l<A.l  .111. 1  <f:i. 
,//.__  //,  //    i-~  tlu     \"^'   .Inc   to 

tlu!  several  hx.iraiilu   nataiuo. 

Ill,  .in-ili.inil).i  thus  CMii- 
.tiiKtrd  Mcni.  to  uui->.-  a  -.toad..!- 
11,,-..  ol  u.iter.  ami  <  \l><'riincnt  >ii.'u- 
tli.it  thf  \ariatir>ii  "I  ^  vcl  .it  ■>  '  i- 
Mii.ni  .111,1  1^  only  .il.out  1  t..  \  mk!.. 
1„  ,„„nj,.  uludi  arc  Kd  -m  l.ulh 
Km    3i.»-  sulrs.  I'i^;.   ^\\.  till- •>trii<liin.s-i  ,>f  the 

U-n.i.   ...nucctin,   the  two  at   tbc  n,>,..r   .n,l    I.   ..    i— ,,.i 


/\,-//ys/^  or  cENTRinc.-ii   I'lMr. 


0:1 


A    \,ilvc    in    till-    inidillr    of  tlu-   pipr    m,i\-  CdiniiuiniL.itc 
1  i-,u  li  I  luniliir  ina\'  i\'     ■  In-  c<  'lit  roll  re  1 


pipi- 

w  itli  ;i  \-aciinni  pump,  .iiul  i.u 
h\-  ,1  -(  parati    \  ah  c 

'111.     wat.r     aiipaiciUlx     ilou  ■.    tlir(iu,L;li    the    bend,  [la-^t    tlu 
•  Millet  ,   a^  o\i.T  an  ela-tK    tii^-liiiMi. 

Tin    A'A;/ r.vr/d'MU    011  tlu    pump  per  --eeond 


wQ 


v„.  u        v„.  u,        wQH,^ 


I  I 


;>ii 


(I  tluretoic 


v„    u        v„  u        gH,, 


(2J 


///,     <fT/(/i  u.  y 


H„ 


RK. 


H„       v„    u        v„  u 


an.: 


gH,, 


—     »/(V„.   u. 


V„.  U  I     1-    llii     tundanii  ntal    i(|uation 


ivivniiiL;  t 


lie  (le-iiL'ii  of  a  i.  enti  ilUL;al   pump 


1  lie  uatei-    s|)rea( 

1.      fo 


•  if    the     pnnip    and,    toi-     -imii 


lilt   luoie  or    le^^   ladiall^    liom    the  eye 
ilieite   ul     i.  alenlation,    it    i-^     often 


n^^^iinv  il  that  r     -  'd' 


h 


'//,  =  ;///., 


am.  ecj. 


hei  oine- 


7/-..    -  \ii,  —  V,"  cot  ti\u. 


.liiih  niav  he  written  in  t' 


le  torm 


«.,• 


;//.       sdl    vJi- 


.t  (i  --^-  I , 


i]\i,idratie  i^ivnv. 


cot  /)' 


gH.,        »  gH„ 


\  '  .'rhJ 


cot  ■  1^ 


»• 


\W  ni 


.f  tiii-  re-uit  the  tojlou  mi;    Table  i 


la-    hicn    pii'' 


iretl  and  L;i"  '  -  '  i'.-  va 


llM-   of 


»     -//,. 


II  risiiondm';  to  ditlerenl 


value*  of 


♦.V 


7/.. 


,ind  /<; 


•t  iiJ, 


V.6 


y ALL  IS  oi  Hi.un:  .;a'(.7./v  rrc 


li-= 


^ 


"  >^' 


"*//: 


I.o 

'» 


3  •'}■'' 3 
3-''33 
3.2()o 

2.(i20 
2.301 

i.i)<rJ 

1-441 

I   •2i>4 


.27; 
.304 

•  3V) 
.3^2 
■434 


2.  lS(| 

2.047  I 
I..)oq  j 

'•775 
1.1.47  I 
1.522 
•■405 
1 .  2')3 

i.i,«S  I 

I  .■>iH> 


■4f7 

.?24 

•  ?'|3 
.607 

.712 
.773 
.842 

•V": 


1-547 
1-477 
1. 401) 

1-344 
;  1. 281 
'  1.220 

I  .  I  I)  I 

1. 105 


.lilS  l.j.io 

.647  1.21)3 

.677  1.257 

.70(j  1.2. "2 


-T-j  1-144  -87^  • 

.773  1. 127  i  .8S7I  : 

.71)5  1. 112  -SrjS  I 

.Sl.<  l.oqS  -910)  I 


.744  I. lis     .S42  '-oSS  •'P2  I 

.7S1  1. 154  I  -'''>('  '•'>f»)  -"JS?;  1 

.S2.>  1. 121  I  .Syo  1.055  .'Ui^j  ' 

.S(ji  l.ixjo     .1)17  ■•"41  •y'"  j  " 

,()()5  l.o?i)     .<M4  I  027  .973  ' 

4=;i  i-o2<)    .1)71  '  "«3  •9'*7,|  I 


/;/,i</i-.--'Vhc  assunipti.-n  of  .1  ra.li.il  tl.w  iVoni  tlu-  c\<-,  i-c-. 
tliat  I-  ')0".  cannr.t  ..f  .,.ur-f  1k'  true  aiul.  i)..-Mbly.  i>  iv.t 
even  ..ppp.xiinatils  ai^cuiatc.  hut  is  solely  made  f-.r  tlir  pur- 
pose of  securin-  siniplicitx  in  tlu'  c.iU  ulatiMU-.  In  fact,  tlu- 
u.it.  I  !1.A\-,  t.'u arils  the  e\.  wuh  a  uiiifoini  motion  parallel  t^. 
till  .i\is  ..f  rot.itinn.  while  at  e\er\-  point  between  the  inlet 
anil  outlet  of  the  wheel  the  motion  of  a  fluiil  particle  i,  th. 
result, uil  of  .1  lon-tint  an-ul.u'  .uee'eration  i.\rt.  21,  thap. 
I,   ,1,1.1    ol  a    railial   .icceleratiou    ,lue  to  eentrifu-.il   force,  vi/... 

/  I-'-. 

At  the  inlet  the  iip,in-!e..  in,i>  he  \arie.l  hetueeii  wide 
limits,  hut  its  v.ilue  .houl.l  he  such  .is  to  m  ike  the  efficiency 
as  ^rcat  as  possible.  Mut  ,f  cannot  be  expresscil  as  a  fimclioti 
of  the  mcch.mical  .i--<l  hydraulic  resistances,  and  it  is  tliercfnro 
impossible  to  find-  .malytically,  the  \.iliie  of  ,r  which  will  make 
these  resistances  a  niinunmn  The  best  v.due  for  «  can  only 
be  determined  by  .1  very  extensive  ser.c  .  of  expc-riments 
Accordint;  to  the  best  practice,  ho\\e\er.  ..  n-u  illy  lies  between 

5u    .\\\i.\  (>o". 

The  ciirvi-  of  til.  hi. id.  i  >  some  cases  a  circular  arc.  In 
many  first-class  wheels  it  is  ,1  cycloid  deve|o|H-d  by  a  circle  of 
a  diameter  equal  to  one  fourth  ol  the  diameter  of  the  wheel, 
the  cycloidal  arc.  at  the  innermost   point,  bein;,;  tani,'entia!  to 


.-^N.-tlYSIS   <•)/■■  CESTRII  I'C.ll    I'i'Mr 


;?7 


the   huh.       I'rix,    aL;.iln, 


li.i^   .k-ihunl    tin-    r.ill.'V.iiu'   t'orniula. 


V,\\nv^    t 


he    ;ui"U'    (''   hctwiHii    tht-    lilailc   ami    tlu    dnx-itidn    of 


rotation  at  any  point  di-itant  /   troin  tin-  axi- 


ot  <!> 


2  n<ii. 


O     > 


2nr 


WlKTC     J  711(1,    --    I  J.T/ 


i//u/-     /' —  Work   '■!    jiuuii)  tM  r.uhu- 


thus  t.i  inner  rn.l  of  hlaiK- ;  7v  =  raihal  \-.hH  ity  at  inht 


fi  "  nuni 


her  Mt  liiadcs;    /  =  tllickncss  of  hladc. 


li 


y  j)h)ttni!. 


thi'  vahic-^  of 


c^nondin!/  til  ditlcri-nt  \ahiL-s 


>rrc>^l 


of  r  tin-  i'ur\(;  of  the  Idadr  may  he  detnu'd. 

It  is  e-isentia!  that  there  sh'udd  he  n<'  di-.--ii)at  i' mi  •  •]  enei\L;y 
in   cdd\-  motion    at   the  iidet,  and  the   dn-eetion    "t   thi-   i.lati\e 

the    hli(K'-tin 


vt 


locity.    /'  .   sliould    theretnri-    h(     tani^ential    t. 


at  ii.   l-\'^.    '>  1 3         1  ln-'ii. 

from  the  tri.iiiL^le  (i,/<.    i\'  =  •\'  -\  u^'    -  2r^/i^  eos  j',         (5) 

nm\  th>;refiire 


l\'  -  /?  ,    /t.r  -  !<;'      ,'•/  —  ^*  _  "v..  i-'"^  '^      ^V'l  cos  ;/ 


J. 


A"  A""  ,C 

Tlie  water  haves  the  ulieel  u  itii    a  \el,uity  :• . .  and  earrii 


(7) 


a\Vii\-,  in    its  enerj;)-  "l    MMtmn.  vi/.,   ^-^.  an    imp-'ilant   i».itiMn 

of  the  uoik  dime  mi  till  |niniii  li\  tip  piiiiie  nio\er.  I(  tile 
whole  of  this  eiierL;>  .  onid  h,  made  avadahle  for  increasing 
tlie  pnnii)in^'  |io\\er.  then,  hy  llennniilli's  tluMri  ni, 

Also. 


^) 


'9) 


th--  t( 


14  '       -   tl'^ 

'    hi'ine    the   variation    nl    pH       iiiediead  due    to 


9 


t::( 


55S 


lOllU:. 


t.:i-iUriUiL;.il  acti.m  iH-tut-rn  the  .\h<-il  ini.t  ami  nntl.t.      Hi-iice, 
1)\-  cqs.  (S  I  ,uu!  en. 

K  +  iL-^  ;\7=     _>.  '-    -   "^     -V 


T,//.,  COS  fi 


,*.' 


ilO) 


^^1,,.,.,,      /,^    ^    //^    _^    //,    +    /';    -    tl"'      ''^-"l      ^-I'li^  ■'''■">      '"     t'"-'      ^"^■'' 

lu'diaiilii    n-i-t.iniT-^. 

U   the    mkl    lluw    i^    i.uhal,    i..-..    it     r^^'p'.    and   if   tli.- 
hy.lrauHc    n-.istaiiLO>    and    ihi'    vclndty    ..t    .lehvcry    can    l)c 

,linntn-h.',i  t..  --ucli  an  .xtc-nt  that  //,  ,uui  \\^  l.r...in.-  .ut'tkiL-iUly 

-A 

Muall    tM    W-    diH. -aided    with.-ul    much    i-rr..i ,  llieii    •'il    (lO) 
heeonus 


//..  --= 


7'.,//,  CiiS  »> 


'.,•    "; 


III) 


„;/,/  ///,•  /,'A;/  ,r.,xilal'h-  ,>u>x_v  is  transh'i  >i,.,{  into  iis</iil  .->•<'//(•. 

;-,,////,  —llic  watiT  i<'<iKs  Irnni  tlic  nutlit-iirfaci-  into  .1 
casing.  «'!•  v..lutv,  whicli  Miimnnds  the  vvlie.^l  and  uhuh  should 
always  In-  ilcsit^ncd  in  such  a  manner  tliat  t!ie  .h.tui  hancc  in 
tlic  fluid  mass  mi-ht  Ik-  as  small  a>  possible,  mikc  the  least  dis- 
turbance in  till-  stream-line  motion  eauses  a  hi-s  ,.t  enerijy  m 
shock.  'I'hus  its  secti..nal  area  on  any  normal  plane,  throu-h 
the  centre  of  the  wheel,  should  be  p, .  .port;.  .n,d,  to  the  .i-iantitv 
of  water  wliicli  (lows  across  the  section  m  the -.ime  .-i\en  Unu  . 
and  the  corresponding'  moan  veloeitx  ofllow,  ;  , ,  in  the  \olute 
is  necessarily  constant.  it"  the  uidth  of  the  volute  is  ,dso  con- 
stant. Its  profile  ^^ill  .Mdentl\  be  an  Archimede.m  spiral.  Wy 
making;  tlie  ^jra.lually  increasing  sections  sufficiently  1.  rg.-  the 
velocity  of  flow.  ,-,.  maybe  m;u:-  very  small,  and  a  bell-moutli 
entrance  into  the  dischar^u-pipe  m.iy  become  unnecessary. 

\'\\^s.  320  to  325  are  of  interest  as  showing  tin  evperimeii- 
tal  stajies  tlirou'.di  which  l-arcot's  pump  pa-sed  :n  the  proec 
i)f  its  ^'radual  devcloi)ment. 


M  ■ 


,-/AV//.  y.s/.S    (J/-    CES IKII  I  ij  II    ri   MH. 


5-9 


Tlu'  .i-^suinption  tliat  tlu-  tut.il  available  cm;r^\-  iiia\-  he 
traii.slnrincil  intn  usLt'ul  xMnk  i'^  altoLjctluT  iiiailniis'^ibk;  in 
practice,  a-J  a  larL;(--  purticui  is  (.oiisuiiucl  in  o\'i.ti  niiiiiv^  frictional 
rc-sistaiKc  and  in  tin-  )>r<iiliu  tion  '<\  eddies. 


Fk;  320. 


Vu..  3^4. 


Fig.  322.  Fii',.  32?. 

Af;ain,  even  with    tin     itwA    ]i(iliitl\    d( -iL^iied    \cilnti.  the 

h\)i"th<M~    that    thi    whiile    111    the  enei\L;\    n|    nmli'in,      ■  ,    jTiay 

-A 

lie    utilized    wi    HuriasinL;    the    ]ir,ni])niL;    ixiuer    is    iiiitciiablf. 

The  \vater,   as  it  leaves  the  \\lu(  1.  witli  ,1  vciocit)'  r, ,  illipin^,'es 

ii|)on   thi-  (liiiii  mass  in   the  \i)hit(  ,  .md   tin.'   radial   CDiiipoiicnt 

;•,     <<(  :  ,  must  nt-cessarily  bi'    .dimist,  it' not  wliollj'.  destroxcil, 

the    I  orrt'spdiidiii':    li>ss    m|    luad  beiili'  .        1  lie  taneential 

tDmponcnt  i>f  ;.,,  viz.,  :„,   ,  is  also  chan^;eii  into  ; , ,  the  velocity 


. '  ■i.\ 

jlr 


5^ 


560  .\/.-;,\7.VrLA/   hlllUi.SCY    Oh    Ct:Sri<ll  i  •'..-IL    I'iMr. 

of  t1i)U   ill   the   Milutc,  anil,  if  tin-   iIi;uiL;t.-  wcvv   :^>-,i//!t,i/,  there 

u.piihl  he  .1  ■'i\\\  of  he, 111  I  liu.il  to     '"       —        ,  hut,  as  the  i  haiiijc 

is  iil'rKtl.  tliere  is  a  llls^  nt  luMil  in  shmk  eiiual  tu  — ~'- — . 

II  Mice   the  lit/  ,L;.iiM  iif  hea'l  a\aiial)le  tor  nureasin^  tlie  pimii)- 
m'^  power 

'        ->(>■  ->.>■  ->  ,r 

->S  -.•>  -.s 

V,      V:,.  V,  1 


(.2) 


whicli  is  a  luaxinunn  and 

I  v„.  V, 

~     2      2g  g 


when    v„ 


2V, 


The  term     '     '"  '    slioiili.1   be    sul>stitiitiil    tor     "'    iti   en. 

I S   ,  ,iiul  then 


lleiiie.    l>y    ( qs.    (7)    and    (i.v,  the    to!louiii;_;   ei|iiation    is 
nt)taineil  instead  ot'eq.   (10): 

-.s  g  .V  .T  2',- 

_  «j^  —  Fj*       ?•,//,  co.s  r 

and  therefore 

,,  U,-  V,'         V.(V„.  V,)  V,,  ,  V  U    COS   I- 

H-  =    ■-        ■  4-  -  h,.  14) 


M.nnnuui  /■  iti,  i,  II,  y.  -  -\\  tile  lei  ms  //,  ,  '  ''.ami 


r'/rv"  —  V,) 


I 


M.-iXIMIWI  FhFICIESCY   (V    Cf-MRIFi'ii/ll    rCMf 


;f)i 


,irc  •^utTuirntly  sinall.  as  compared  wirli  //,  ,  t"  he  (lisrcL;ai"<l('l 
vutliDut  imiLh  crnir.  .\nA  if  r  r-  yo  ,  tluii  vt\.  <  141  tjudiiio 


H„ 


u;       V^ 

2g 


<i5) 


I'.iit 

I  licrcfiiic 


sin  f? 
sin  (Ji  +7)' 


sin  /i  cos  S 

"^  sin  (^A  +  rfj' 


J  sin=  rf       /   _     .,  sirw/sin  (/3+ 2rf) 

"■'•''    "  ""  "''^  '  ~  sin^V/TTrf) »   ~  "'"        sin^  {Ji^S)     ' 


11  n  '^in-  (fi  -\-  'i') 

2^'//„   "  sin  fi  sin  (/»  +  2fJ)' 


(i6> 


nid    flir    crfKlcnc\'    /;  = 


^//.    _    1        sill  (/'y-f  2^5) 
f.„"Wj       2  tns  f)'  sin  yfi  -^  rf)' 


;;  =     ;i  -j  tan  •    cot  ili  -\-  ^)\. 


'I"lu;  efficiency  iiR leases  as  fi,  tlic  ontict-tip  anisic,  dimin- 
isiies,  and  would  he  iniit\-,  i.e.,  perfect,  if  li  could  lie  O 

If  tile  blade  is  radial  at  tin-  ..utlct,  i.e.,  if  /i  =:  go  .  then 

the  ifTiciencv  =     )  I  -f  tan  '?  cot  190"  +  "^tl 

I 
-^-  ^(  I  —  tan-'  rfj, 


<uid  c'lild  neviT  exceed  ^. 

I'll!  an\-  L^iven  value  of  fi  the  cffu  iency  !<;  a  maximum  when 

/?-|-  2''         90   . 


I^H' 

i 

'1:1 ; 

i  '' 

5')J  M.-I.XJMiM   Hlil'Jl.NL)     Ol-    (:i:S'rKU-L\,.-il.    t'l.MI'. 


{3 


'<A 


l=M 


'I'lii-.  can  !)(.•  ca<il_\'  sluivsii  ,m:ilytically,    <  iv  i^ciiiiuti'ii  all_\-  as 
tdlK.us; 

L  pi  in  any  line  .  I  />  a-.  (iianutiT  dcscrihi'  a  scinirii\'U  .      1  'imw 

a   cli'ii'l   .  1 1,     making;   the   aiv^lc    li 
''  uitli    .1/''.       Draw    an_\-   elmni    .!/> 

niakiiiL;'  an   ar._L;le    -    with  .  I  ( ',  and 
join  /'/.'  intcrsL'ctiiv^'  .U    in  ( '. 
/■;..        ,-''    "^   ^"Ov     \  1  he  etVicienc}'  i  ;  L;reMtest  \\!ien 

tan  '>'  cut  { IS    :    "  -    has   -ts   •/reat^•-^t 


vahie. 


circle  is  a  ri"ht  an''le. 


tan  ')  Cut  I  I)  -'    •'> 


Hut    ^ince   the  ani^le  in  a  seni 


an.l  th.-  et'tu  ienc)  is  therelnre  i^ivatest  wlun  i-  a  niaxiinuni. 

\,,\\       '  ,  is  nil  lintli  when  />  coincides  with  ./  and  aK.iwith 

i  \  and  must  C()nsei[uently  he  a  maxiniuni,  or  stationary,  at 
some  position  ol/^  between  J  .ind  (  .  This  i)ositi()n  is  at  once 
founi!  trohi  tlie  condition  that  i(  /',  is  a  consecutive  point  aiitl 
if /',/'' is  joined  intersectin;^  ./(in  ( ',  .  then 

PI!  ''^  i\n  • 

sM  thai  /'/',  must  1)C  parallel  to  ( '( ',  or  .IC.  and  is  th<'i  fore 
a  tan;,a'nt  to  the  semicircle  at  /K  which  is  nece--s,iril\-  tin-  middle 
point  of  the  arc  .W.      Hence,  since  the  arc  .  //'  ~  the  arc  C  /), 

the  anj,'le  J/.7>  ^  the  ani^le  (  .1/)  =  <j, 

and  theretejri; 

fjO""  -  (fi  J    rf)  =  rf, 

or  fi  +  2rf  =-   90°. 


MAXiMCM  liii(Ji:\<:y  oi    ci-xTRirrc.-ti.  rcMi:  56^ 

I  IflU  t',    ti  10, 

the  max.  cftKifiK-)-         1    •-  tan  o"  cot  kjo'  —  rtj 

'1  !ic  outkt  \-cl:)citic-;  i.-ni-rc^iKiii(liiiL;  t"  this  inaxinuim  i-ffi- 
eiciK\-  ar.-  rcprcsciitcil  by  tlic  >iik's  .,1'  the  triaiiL'lc  .'W.  Fi<;. 
^2j.     1  he  two  triant^flcs  />//•  ami  /a-/'  arc  ccpiai  in  c\cr>-  rcspeL't. 


Ku;.  327. 
-Al-io,  2  sill-  ff  =   I  —  cos   jf)'  —  1  _.  sin  /^ 

and  J  cos-  f?  ^    I  -1-  cos  j,?  =  i  -i-  sin  /y. 

1  U  111  f 


ir   _   .  «'"  '>  ,  /I  -  sin   A* 

-sni  100    — -  It 


-\i  -i-  sin  li)  ' 


and 


-   ,r  i       I-.  ,    -  Sin  fi 

'  -  -   I  -J   Mn  li 


\    ■   sin  n       ,. 


n 

!  ..:  >:,! 

1 

564         MAXiMLM  iifi(:ii:\(y  fv   ci  \ri'iiiu,.-u  ri  .\u\ 


Alsc.    ,-.,  =  «.,-. 


sin  /i 


=  /^  ,  =  »  Mil/'  .  -,-,''',  ■ 


sin  or  -\-  6)  -  co-  ')' 

/-.in  a  (  I  4- sin  /^) 
^=  r'.,  cos  fJ  -"-  ?r,  sin  /:<      *  /  -_^ 


2^1  f.,  , 


Mi:^ 


I!::   «■    '-,!      il 


s  ir, 


? 


and 


/s[n  /;;n  —  sin  /:;)      ,^ 


l''rnni    tin -,c   ciiiatinns   the   fnllowini;-   Tahlc    li.is  \)ccn   ]>re- 
p.iicd  : 


"1 

'a 

3 

i 

♦'v^'/a 

♦'«//. 

.20 

1 

.92 

5* 

42-  30' 

2.4'(7 

.  2()(: 

8  26' 

40  47 

1-977 

•3 '3 

•25 

.87 

10 

4<) 

1.S33 

.41'. 

.267 

.So 

J5 

37  30 

1.56 

.^oo 

.30 

•79 

20 

35 

1.40 

■53 

1   -33 

•  74 

30 

3<j 

1.22 

.70 

■35 

.67 

40 

25 

1. 13 

.So 

•34 

.(>! 

45 

22  30 

1.09 

.R4 

•  .12 

.s.s 

5'> 

21) 

1.073 

•!'7 

.30 

.56 

■;S  -,6 

15  42 

1.041 

.92 

•  25 

•  54 

60 

15 

1.038 

•93 

•  24 

•53 

t 

fi 


II 


Hi'       '^ 


Tlic  value  of  the  raili.il  i  oni|)<  nu-nt .  :  ,  ,  is  'greatest  ulicn 
sin  /'(i  —  sin  fi  <  is  a  niaxununi,  i.e..  \^lieii  sin  il  -■  A  .)r 
fi  ^  30"  ^  rt". 

Two  values  of //',  tlie  one  less  and  tlie  otlier  ;4rt  iter  tlian 
30°,  corrcspomi  tn  e\ei}'  other  \ahie  ol  r^". 

(ieiierall\'  speakin;.:.  •".  '''-'^  l)et\\een  ^»  2,i,'-//,  and  ',  »  -X/^.,- 
and  tlie  assiiinption  i--  sduuti'iies  made  that  the  r.idial  eoni- 
l)(iiu-nt  (if  the  \el<ieit\-  i>\'  the  u.itir  as  it  passc  -  tliioii-h  the 
wlieel  is  constant  and  e(|iial  to  ',  ♦  2^//,. 

M.VAMlM.i:. — A  contrit'tij^'al  piini]),  \\:tli  an  outlet-tip  aiii^Ie 
(/I)  of  20 \  has  an  efficiency  of  60  per  cent.  Assuinin;; 
r/'  =  i  *'2,;'//, ,  then 


IVHIRI.  rOOI.-(  HAMIiF.R. 
gH^  '--  .61  .,'■„."  ~  .C'if.ui.  —  7',    cot  20") 


5^-5 


and 


=  .C'Kl^r,  -  ^  t  Ji--/^  X  2:7475), 
7/,,  r^  1.33  ♦  J^-7/,. 


Al 


v.,  =  rv"  coscc  20"  z-      \  2i^-f[^^  X  2.9238 
4 


:^.  \  2,v//„  X  -/"SO'JS- 
Tlicrifiirc 

71.:  —  /',■  =--  1 .20S  y  2i,'-/^,.- 
Hcnco,  if  the  inlet  flow  i^  r.idial,  c(|u,iti(>-i  (  14)  ^nvcs 


//,.-  -  I'r 


1.208//,, 


and 


A    4-^"-^^^^^-^^>=.20S// 


Ccrtniii    cxi'-tinL;'    cxjicrimental    ii'-^iilt^    i,'i^e    ,42//,    as    an 
avcra;.^e  \.iluc  nf //,;   and  taking  .03//,  a>  the  .ixeiagc  value  nf 


■ ./ 


.  then 


=  .42//.   -;    .r,3//,  -  .20S//,  -r.  .J42//,. 


I  he  term niiwt    necessariU'  \ar\'  eonsiderahlv- 

.'.'■  ... 

witli  the  ile<iL,Mi  iifth'-  pnnip 

3.  Thomson's  Vortex  or  Whirlpool-chamber     V'v^'^.   32S 

and    ;\:i)).  —  It  lias   luen    su;;:4esteil  that  the    inerL;-y  of  motion 

inherent    in    thf    water,  a-    it    lea\'es   the    w  luc  1.  max    he    more 

eom]>letel_\-    utili/eil,     and    the    pumpmi;    jiowcr    therefore    in- 

iri;a-(d,  Sy  the   ,iddition  ot   an  I'xterior   ehaniher   of  radius  /-.j, 

in    which    tlie    water,    in    virtue   of    its    motion,    is    left    free   tt> 


5 


X 


fe::? 


566 


iinn<i.i'i)oi-i:n.-t\iHHR. 


revolve,  ami  teiuls  to  assume  the  condition  dcsirrnatcd  by 
James  Thoiiison  as  the  vortex,  or  \\hiilpool,  of  free  mol)iIit} 
The  centrifiiL^al  aetioTi  of  this  lliiid  ina>s  develop-,  an  <iiit\\aid 
force  which  is  added  to  the  outward  force  developed  within  the 
wheel  ami  materiallx-  increases  the  ])umpinL,'  power.  llie  out- 
ward   force    produced   within    the   wheel   is    due   to  centrifugal 


V\r..    328 


action  onl\-,  if  the  blades  are  radi.d  ;  but  if,  as  is  n-cncralh"  the 
case,  the  bl.id.es  aie  cur\ed,  it  is  p.irtl}'  i.\ui:  to  the  radial  (.om- 
p.  lUeiit  I  if  the  jires^ure  between  the  bladi-s  and  the  w'.iter,  and 
this  pressure-  nuiy  be  \  ere  ;-;re.it  it  tin  jmni])  is  run  at  a  hi.L^h 
S[)eed. 

The  chief  pi'opertie^  ch.iracteri/ini;  the  tluid  m.iss  in  tlio 
whiiln'iol-chamber  ;'re  the  fnlhiwinL;: 

(I  I  I'"..K  h  tluid  ]).u'ticle  niove-^  with  a  \-tdocit\  (.")  in\ersel_\- 
proportion, il  tn  it--  di>t.ince  iri  from  the  ,ixis  nf  rotati.  in.      Thus 

'■.'■.  =  '••■  =  '-.J':, . 
T'.  beinif  till-  xvater's  velocity  .it  the    outle'-surf.u  e  of  the  whivl- 
pool-chand)er. 

(2 '  The  anLjle  ("i  between  the  radi.d  d.i'-t.mce  (;-)to  an_\- 
p.article  and  its  d,irecti'>n  ^f  motion  i- cun^tant ,  .md  the  stream- 
lines are  therefore  ecjuianj^ular  spir.ds. 

Thus  if  rv'  '  ''ind  ."■-,,'  are  the  radial  ,ind  tangential  coni- 
jionents  of  i'^ . 

','  =  f,  cos  rf,      rv       =  ;•   cos  rf, 


T',„    -  J'2  sin  rt, 


r',  sin  (?, 


1     !' 


liillK/l  OOL-CH.-IMHf-R. 


y'l 


and  tlicrcfiirc 


2  '  :i  


(3j  l^.K'h  jiaitiLlc  is  free  to  iiha  r  t..  ,in_\-  jxi-itidii  within  the 
wliirlpdol  without  interfering;  with  ihu  i^cnc-ral  nintion  <il  the 
other  ])artick'>,  as,  in  ni(iviii.i;  tou.ini-  (^r  Irum  the  iLntrc,  it 
assumes  (if  itself,  sui)iect  simp!)"  to  tlie  laws  cf  motion  iimler  a 
eeiUral  toree,   the  velocit)' due  to  it>  po-ition  m  the  whirlpool. 

(41  I"<ir  an>-  e(iual  particles,  whate\er  positions  tlie\'  may 
momentarily  oecup>-  in  the  whirlpool,  the  ■  um  of  the  eneri,nes 
corresponding'  to  velocity,  to  pressure,  .ind  to  heiylit  i^  con- 
stant. 

Thus  each  jiarticlo  L;ivcs  up  its  velocite  in  accordance  \vith 
the  law  .if  motion  just  stated,  and  the  head  available  tor 
increasing'  the  i)umi)ini;  power 


-\"ain     the    term    '■'^-" ^,    representing;   the   i^ain   of 

head    in    passin;,;    from    the    whirlpool-chamber   into  tlv  \dlute, 
must  be  suli^tituted  fir  the  ttrin     '     "'- 


the  efliciencv 


Thus 


in  which   //.  =   u.rr,,."  —  ttju.,—   /',  cos  0).  .and   the   .ictu.d   lift 


u--V,=      v„-Y       r.'\v,(v„.'    -V,)   ,       v,r      , 

H  -    I -,    -  H —  br- 

2g  2g  r,//  g  2g 

Kx,   .Assume  tliat  the  (our  List  terms  in  tiic   prrcefHii;-;  equation  are 
siiiricienlly  small  to  be  (iisregardeil.     Then 


the  efliciencv 


2u-i{u-,  —  y-i  cos  fl) 


■^ 


1 

til 

s 

h 

1 

1 

568 


H-HIRI.R,  n  U   CH.-tMHHff. 


I 


$ 


Fi)  ./.   i.et  li        >)»■.  I.e..  li-t  tlK-  bhuic-  uulli  t-lip  b<!  radial.     Then 

7'a"  =  ;<i      and     v,"  ~    I'j. 
Tlicrcfcjre 

«.'  -  ir  f  (//.'  +  r/  (i  -  '■']) 

the  oii'ii  ii'iicy  7_  . ''  ''   I 

~"/ 

_       _    I  >/'        I     r./ri" 

-  '  1'  ~  :    ;</    ?  J-  ■ 

SeiOtii/.    L"t  fi        u.  i.e..  let  the  tjl.ide  ..ullet-lii)  l)e  taiif,'enl  ial.     Tlien 
?/„"  =  ttj  —  r,      and     7v"  =  o. 

Ti'<Tef..r(; 

«/  -  r,»  +  («,  -  r,)«f  I  _  '■'I-) 

the  ellieieii.  y  = \         ^»   / 

I   '  ,'         1     /'»/-,' 

=  >  —  ,    +  — .. 

.:  »>-        J   «•  ri' 

Till-  ililkrciuc  liftwccn  '.lic-c  two  efficiencies  i.s  coinpara- 
tivcl_\-  small  and  (iiminisJR's  a-,  tlu'  iliaiiictcr  of  tlir  wIuiIim...!- 
clianniiT  iiu  rcases.  IIciico  tiii-ir  values  arc  imt  lar-dy  iiilln- 
cnce<l  ii\-  tin-  .iiiL^lf  /)'. 

'1  he   .iIka'c    1i\  iintlii  ti.  .il    tiuory   si  eiii>   'o    indicate   tli.it   a 


Fi(!.  330. 
wliirlpool-chamber  adds  to  the  efficiency  of  a  centrifu^'al  pump. 
Opinions,  however,  differ  widelv  ,is  to  tlie  real  ch.iracter  of  the 


PR/ICTICAI.   COFhTlCIENTS. 


5^9 


flow  ,,f  the  water  witliin  the  piinip,  an,!  as  t<.  tlie  loss  cif  eiirr-\- 
in  shock  on  entcrin-  the  u  hirijiool-clianiher  <,r  the  vohit'I-. 
Sonic  eminent  authonti.s  atlxocate  a  ,t;raduall>-  chininisliin- 
section,  I-j-.  330,  on  the  -round  that  it  teiuls 'to  pr.nhice  a 
stra.l.er  action,  wliile  otlier  anthorhu  ~,  equall)  eninunt,  chum 
tliat  a  Harm;;  vortex  ten.is  to  increase  tlie  etricienc)  .  and  it  is 
ur-ed  that  by  widenin-  the  chaml)er  from  the  depth  at  tlie 
wheid-outlet  to  a  mncli  -reatcr  depth  at  its  exterior  surface, 
the  uater  uill  lose  its  ener- v  of  ni.  ,tion  „uu  h  more  rapidl>- and 
u:ll  leave  the  chanilnr  u  ith  a  \ .  hn  it.v  more  nearly  cjiial  to 
that  m  the  di-.cliarL;c-pip(\ 

I'lvperiments  are  ur-ently  needed  to  throw  li^^ht  upon  thi-, 
important  suhjei.  t. 

4.  Practical  Values.  -Let  ,/,  ,  7,  he  th,'  .hpths  ,,i  th.-  inh't- 

a!id  outlet-surla;i'^. 

Let  /,  ,  /_,  he  the  l.lade  tim  kni'--,  .it  mlci  ..nd  outlet. 
Let  //  he  the  iiiunlni    ot   1)1. ides. 

1     t  the  iidet  .irca  =  section.!]  .ire.i  of  ^upjjK -pipe. 
i  heil.   it   f         90' 


ro 


=  {2  7T/:,  —  ///j  cnscc  ft\/.;\". 


the  c.efCuient    ^^    luin;:   an  avcra^je  value  and  d,  pending  i.po-i 
practic.il  loiisiderations. 

The  following;  vahies  are  >om.  tmiev  .adopted  in  practice: 


r,  =  2/-, ; 


«  =  4  to   10; 

'1  =  ft  =  .2  in.  to  .625  in.  ; 
.;./,    --6/-,: 

r,'V//„=  .661  f; 

^t  ~  ''1  ">■  =  i'/, .   according   as  the  pump- 
faces  are  parallel  or  coned. 


S70 


I'x.iMri.i; 


V-<  Th.-  hypotlietical  adva.uaKO  »(  h  wliirlpool-chanibcr  may  hr 
observed  bv  a  c.nsi.lcration  of  the  conr.aralivc  etikienct-s  of  two 
rnimps.  wlii'ch  arc  precisely  similar  m  every  respect  exceplmj,'  that  one 
has  a  whirlpool-chamber  of  62  ins.  diameler.  Each  pump  delivers  20 
cii  ft  of  water  at  a  speed  of  22;  revolutions  per  minute.  The  diameters 
of"the  suction-  and  dis.har-e-pipes  =  20  ins. ;  the  diameter  of  the 
wheel  =  36  ins. ;  tlic  <iepth  of  the  whirlpool-chamber  =  the  depth  of  the 
wheel  at  outlet  =  s\  i"S. :  y  ^ ')o' -^  ft  =  4^°  20'  22':6y.  number  of  wheel- 
blades  =  6;  thickness  of  bhide  =  t  in. 

The  actual  lift  is  K'vcn  by 


5 


?*, 


N 


or 


3g  2i'\  '"/ 


v./' 


-    h. 


accordin-  as  the  pump  lias  not  or  h.is  a  wl  irlpool-chanibcr. 

«,  =  ^^  =  J5m  f'-  per  sec; 

cosec  /I  --  1.484: 

-  5.523  ft.  per  sec. 


144  X  20 

■        \  "  ~     7  ' 

.9  ^  ff  X   36  —  6  X  1^;    X    l.4«4  (  5« 


"•  I  ■'  '-  J '^  s 

r,  =  Vr'  cosec  fi  =  S.196  ft.  per  sec; 
»,t-tV_  59" -473. 

rot  fi  =  1.097; 

T „."  =  u,  -  Vr'  col  fi  =  29.298  ft.  per  sec; 

;.  "  —  ^  O;'-  ft    i"T  ••'■'.: 


v."       vr  +  »'«.'"=  8S8.876- 


i^.iin  of  head  in  |)assing  from  wheel  into  voUite; 


!,,•/         r?\      888.87^./.        i8»\  _  294.596. 

2,1' -.v)=    ^<    l'~3.')-"T   ■ 


F.x.-isu'u:. 


57> 


l8 


!9.29S  =  T  7.01 2  ft.  per  sec. 


U .V  ■      r=    1035.889. 

lleiicc  lor  ilic  puiui)  -i-itlioiit  a  wliirlpoul-chambf r 

59 1. 47. >    ,    184.50*5      41-986 
i'  ^  .<.' 


or 
and 


the  ofllciency 


^//a  =  733995 —.4'^'-. 

A'/''^     _  733-995  -m'''  .-    ,oc  _       -C^- 


.708 


1035.889 


/'ii-,./  1035.889 

ir  llio  pump  rr///;  a  wliirlpool-iliiiiiihcr 

//         S'J'-473  ^  =94-59<'     ,    7I-9I9        4i  9^6       1 

=  9«6.oo2   _^^^(,8.46-/,,,.t.. 


9ifi.oo2  —  »,'i4. 

It)  13.889 


1035.889' 


wlncli  is  coii-'inlci.iijly  greater  tli.iii  Uic  lii--t  illiciciu  y- 


)    ;    ;>l,ia| 


liX.iMrms. 


EXAM  I'l-FS. 


5 


hi 


I.  I'iiiil  tlie  H.P.  ri-qtiirc(i  to  drive  a  centrifiifial  pump  of  14  ft. 
fiiaiiictor,  aiifl  with  radial  vanes,  making  60  rcvoliitiotis  per  niiiuilt-  and 
(Icliviiini;  900,000  j;allons  nf  watci  per  lion  r.  If  the  lift  is  30J  ft.  find 
the  1  tFuicncy.  Assume  ll.,»t  the  .vater  nn  cnterinj;  has  no  velocity  of 
whirl.  '  '  -'"f-    ^7-5;  -S- 

1.  The  wheel  of  a  rentrifuf^al  pump  is  .6  ft.  in  diameter  ;  tlie  turning 
moment  on  the  spindle  is  12  Ibs.-ft.  If  160  gallons  of  water  are  r.iised 
per  minute,  find  the  mean  veloeity  with  which  the  water  leaves  the 
wheel  ;  assuming'  that  on  entering  it  h.is  no  vcloi  ity  of  whirl. 

.  Ins.  24. 1  ft.  pt T  sec. 
.  centrifugal  pump  has  a  3i>-iii.  wheel  uf  a  uniform  hreadth  of 
J  I  ...  The  whei'l  makes  225  revolutions  perminuteand  delivers  co  cu. 
ft.  of  water  per  second  into  a  discliarge-pipe  of  20  ins.  diameter.  The 
angle  id)  of  the  blades  at  the  outer  periphery  is  42'  20'.  .Assuming  the 
velocity  of  dis(  h.irge  to  be  the  same  as  '.he  mean  velocity  of  flow  in  the 
volute  and  disregarding  vane-thi(  kiiess,  Imd  (.i)  the  peripheral  speed  ; 
(/')  the  vetocitv  of  whirl  and  radial  velocity  of  flow  ;  ir)  the  g.uii  of  head 
available  for  u>efid  work  on  entering  the  volute,  and  (./)  the  ellicieiicy. 

There  arc  si.\  i-in.  tilades.  If  a  Co  in.  whirlpool-chamber  is  added,  find 
the  gain  of  head  available  for  useful  work.(!i  due  to  chamber;  (/j  on  eiiler- 
iug  volute. 

.;«.(.(./)  33.35  ft.  j>er  sec. ;  (Y'l  20.425  and  5  4  ft.  per  sec;  it)   5.8 
ft.;  «</)  .70S;  (,•)  9.27  ft.;  (/)  2.27  ft. 

4.  .\  ciiurifugal  piinij)  with  a  12-in.  f.in  delivers  itxx)  gallons  per 
minute,  iheactii.d  lift  being  20  ft.  and  xXw f; ross  lift  (allowing  foi  friction, 

etc.)  30  ft.     l"iiidthe  revohitiousi 

.■im.  836.52, 

5.  In  a  centrifugal  pump  tlie  extern. d  diameter  of  the  f.in  is  2  ft.,  the 
intern. il  1  ft.,  .iiid  the  de]>th  6  in.  Determine  the  speed  and  etficiency 
of  the  i)ump  when  delivering  2000  cu,  ft.  per  minute  against  .1  pressure 
head  of  '14  ft.,  the  inclination  of  the  wheel-vanes  at  outlet-surface  being 
90°,  a. id  y  being  also<(</.  .Ins.  6i(). 24  revols.  per  min.;  .4866. 

6.  A  centrifug.d  pump  delivers  1500  gallons  per  minute.  Fan,  16  in, 
diaineier  ;  lift,  25  ft.;  inclination  of  va:  cs  at  outer  periphery  to  the 
tangent.  30°.  Find  the  bre.idtli  at  the  outer  periphery,  niul  also 
the  revolutions  per  minuti-.  assuming  the  jfrM.i   lift  V,  l.e  li  .imes  the 

actual  lift,  and  that  v..." 


if  the  piini])  per  minute  iv^J'  =    -  j. 


i:x.iMrtJ:S. 


573 


Also  fin'i  the  i)roi)er  sectional  ana  <>i  the  chaiiibc-r  surrouiulin^  tlio 
fan  for  the  proposed  lielivory  and  lift.  Kxaniine  the  working  of  the 
pump  at  a  lift  of  i  5  ft.  (r'„"  =  o). 

.Uis.  Hreadlh.  J  in.  ;   revolutions,  700;  23.5  sq.  ins. 

7.  For  a  liivcn  discharge  (O)  and  head  y//).  and  considerinj.;  only  the 
losses  of  head  dne  to  flow  and  to  the  resir.tance  in  the  wheel,  show  that 
the  maximum  etliciency  uf  a  centrifugal  pump  of  diameter  U  is 


A 


o 


A  being  a  constant  flepending  on  the  size  of  the  wheel. 

8.  A  centi  ifugal  pump  with  an  eflii  ieiiry  of  .75  an<l  a  radial  flow  at 
iniet.  lifts  35  cu.  ft  of  water  per  second  a  height  of  20  ft.  At  the  outer 
periphery  the  vanc-angle  i(i)  is  15'  and  the  radial  velocity  is  5  ft.  per 
second.  If  the  wheel  makes  140  revolutions  per  miimte,  find  Ut)  its 
•.Iiameter.  If  the  diametei  of  the  outer  petiphery  of  the  wheel  is  three 
littles  that  of  the  inner  periphery  and  if  tlie  radial  velocity  at  the  latter 
IS  8  ft.  per  second,  find  1/')  the  vane-angle  at  the  inner  periphery  anil  (<  1 
the  depths  of  th';  wheel  at  the  inner  and  outer  peripheries. 

Alts.  (,i)  ;..'55  ft.  ;  (/-)  30'  58' ;  (c)   .765  ft- ;  -41  ft- 

9.  The  pump  in  the  preceding  example  is  supplied  with  a  vortcx- 
cliamt)er  of  (<l  It.  diameter.  Show  that  the  "gain  of  head  "  is  a  m.ixi- 
miini  when  the  velocity  of  flow  in  the  v.lute  is  8. 46  ft.  per  second.  Also 
show  that  the  frictioiial  loss  of  head  is  4-'8575  f'- 

10.  In  a  centrifugal  pump  the  diam<-ler  of  the  fan  =  u  ins.,  the 
depth  -  2  ins.,  the  lift  =  25  ft.,  ami  the  <Ulivery  =  300 cu.  ft.  per  minute. 
Determine  (./)  the  specrl  ;  ((*)  the  etricicncy  ;  and  (< )  the  power  expended 
when  tin.  vane-angle  1^)  at  the  outer  periphery  is  (1)90°;  (2)  45°:  and 

(\)  30°;  y  being  '/i°. 

.•i«f.  (1)  (<i)785     revols.  pel  tnin.;  (/«!  .47  ;  ((•)  30      111'.. 

(2)1.1)805.8      (i«)  .58;  (O  24.4  HI'. ; 

(3)  (,1)846.1       "  (/')  .68;  (c)  22.9  I  I.I'. 

11.  A  centrifugal  pump  delivers  10,000  gallons  per  minute.  The 
actual  lift  is  50  ft.  The  radial  velority  at  th<-  outlet-surface  is  one  eighth 
of  th.it  due  to  the  actual  lift  and  /<,  =  2v.J'.  Find  (n)  the  radius  of  the 
wheel;  (/'i  the    vanc-angles  ;  (.)  the   speed  of   the   wheel;    (,/)  the   efTi- 

ciency.  .aking  y  =  90°;  and  </,  =  d,  ^  J. 

Ans.   {a)  1.9  ft.;  (*)  56°  X'';  23°  '<»':  '<•  33'  revols  p- r  min. ;  (</).74. 

12.  The  internal  and  external  diameters  of  the  fan  of  a  centrifugal 
pump  with  radial  flow  at  inlet  are  9  ins.  and  18  ins.,  respectively  ;  the 
depth  is  6  ins.,  and  it  passes  400  ru.  ft.  per  minute  .igainst  a  pressure 
head  of  16  ft.  The  inclination  </i)  of  the  tlischarging-l^'s  of  the  t.in 
being  ^o,  determine  (,»)  the  speed;  (/)  the  efTlciency  ;  (el  the  power  ex- 


v\m\ 


!  ,   I  i 


i    h 


s 


fc4 


X 


574  FX.-IMPIHS. 

peiulcd  ;  and  (</)  the  inclination  of  the  receiving-lips  of  the  fan.  Find 
((•)  the  ftririciu  y  when  a  whirlpool-chamber  of  36  ins.  di, muter  .sur- 
rounds   he  fan. 

.//;...   ,j)4l,V5.S   rev.)L.    ]rt  niin.  .  f/l    .571;  (.J^i.:-;  !I,P.;  (</) 
19'  IJ'  ;  (f)  .J'Sl. 

13.  The  lift  of  a  centrifugal  pump  is  24i  ft.  Tlie  etririency  of  tile 
pump  is  .75,  and  the  radial  velocity  of  How  at  outlet-surface  "f  fan  is  5  ft. 
per  second.     If  cot  1-  =  4.  hnd  the  peripheral  speed  of  the  fan. 

Also  find  its  diameter,  if  the  f;in  makes  160  revolutions  per  m'  lute 
H'J  =  01.     Kuid  the  loss  of  head  in  hydraulic  friction. 

.his.  44  ft.  per  sec.  ;  s\  ''•  '•  jij  ft. 

14.  The  reciprocal  of  the  efViciency  of  a  C.  P.  is  1.61.  the  peripheral 
(«,)  and  radial  kv")  velocities  at  outlet  are  35  an<l  9  ft.  per  second 
respectively.      Find  the  lift  and  the  vane-an-le  tfi)  at  outlet. 

A)is.  15?  ft.;  tan-'  5. 

13.  A  centrifugal  pump  with  a  >;ross  lift  of  17  ft.  delivers  25  cu.  ft.  of 
water  per  second.  .\t  the  outer  periphery  the  vane-.ini;le  is  So^  and  the 
radial  velocity  is  5  ft.  per  second.  Tlie  diameters  of  the  outer  and  inm-r 
peripherics  of  the  disc  are  54  ins.  and  18  ins.  resiiectivcly,  and  the  liy- 
draidic  elhciency  is  .73.  Find  (,;)  the  speed  of  the  fan;  (/))  the  vane- 
angle  at  the  inlet  periphery  ;  {<-)  the  velocity  of  whirl  at  the  oullel  ;  (./) 
the  diameter  of  the  volute  ;  (<•)  the  diameter  of  the  suction-pipe. 

If  thi're  are  six  J-in.  vanos,  lind  (/)  the  uiiith  of  tiie  di^^cat  the  outer 
an<l  imur  peripheries. 

Assuming  the  velocity  of  flow  in  the  dis(har.ue-pi|>c  to  be  4  ft.  per 
second,  show  that  there  is  a  loss  of  5.026  ft.  of  head  due  to  liylraulic 
friction. 

.his.  (,t)  1 16  revolutions  per  niinuie  ;  (/)  41 '  14';  (< )  26.49  ft.  per 
sccoiul ;  (</)  1.094  ft,;  («•(  33.8  in. ;  (/)  964  '•"*• :  ■i-^  '"«• 

16,  The  vane  of  a  centrifugal  pump  or  turbine  is  the  involute  of  a 
circle  concentric  with  the  pump  circumference.     Show  that  l\  =  I',  in 

.111  I.  F   or  (  )    I'.,  and     /  —     '  in  an  A.  F, 

17.  If  the  lips  of  the  pump-vanes  are  railial,  show  that  the  othciency 
cannot  .-.xceed  .3,  but  liiat  it  niighl  be  increased  to  ,875  by  the  addition 
of  a  \\hirl|iool-<  hambei,  _ 

iS.  A  (cntrifugal  pump  with  a  21-in,  fan  pumps  110  \  3  cu  ft.  per 
second  to  a  height  of  31}  ft.  The  outlet-lip  makes  an  angle  of  60  with 
the  |ieripliery.  The  d<ptli  of  the  fan  is  6  ins.  Find  llie  peripheral  speed, 
the  11,1'.  and  the  speed  of  the  pump  in  ri-vols.  per  minute. 

.\lso  lind  the  loss  of  heal  due  to  fiictnmal  resistance. 

.his.  6c  ft.  per  second  ;   16:3!  H.P.;  654,",  ,  31)  ft. 

19.  .\  centrifugal  piintp,  with  six  i-in.  bl.ides,  makes  140  revoluti-ns 
per  minul'.'  and  raises  jcViJi  tons  of  water  per  hour  to  the  height  of  20 
f,.,  .     I'u,.  I,i  .,|,'..itii;l.-  .itid  r.iiH.il  veliH  itv  of  (low  at  outlet  are  cof'  4  anil 


FX.4MPl.FS. 


575 


21.   In  a  centrifugal 
liar(;e-[)ipcs  =  48  ins. 

:  I    in.; 


'.liickness 


the 


3  ft.  pt'i  second,  respectively,  and  the  iiydraiilic  efiiciciu  y  <<i  tlu'  piunj) 
IS  .1  little  more  than  60  per  cent  (=  j^).  The  wht-e!  is  surrounded  \,\-  ,i 
v()rte.\-(  h.miliiM  li.iving  a  diameter  20  per  cent  f4rcater  than  that  c.f  the 
wheel.  Assnininj;  that  tiie  iidet-flow  is  r.idial.  and  th.it  27'.  =  r  „,"  ,  and 
disregardintj  frictiDiial  resistances,  determine  the  peripheral  speed,  diam- 
eter and  breadth  "f  the  wheel,  and  the  yains  of  energy  in  ft. -lbs.  in  the 
vertex-chamber  .im!  in  the  \'i!ute. 

.///A.  44  ft.  per  sec;  6  ft.:  .S.  17  ins;  8070,  8789. 
20.  Compare  the  eliicieii-ies  of  two  centrifugal  pumps,  which  are 
.irecisely  similar  in  every  respect,  exceptiiij;  tiiat  one  has  a  whirlpool 
chamber  of  48  ins.  diameter.  Each  pump  delivers  ?o  cu.  ft.  per  second 
at  a  speed  of  225  revolutions  per  minute.  The  diameters  of  the  dis- 
charge- and  suLtliin-pipes  —  20  ins. :  the  diameter  of  the  wheel  =:  36  ins.; 
the  depth  of  the  wheel  and  the  whirlpool-chamber  at  outlet  =  3^  ins.; 
;'  =  90°;  /J  =  22  38':  the  number  cjf  wheel-blades  =  6;  the  biade- 
tliii  kness  =  ^  in.  ;  /ir  =  .3//... 

v/«i.  .73  —  A  and  .79  —  ./  where  .1  —  ■'"    '. 

5U3-6 
pump  the  diameters  of  the  suction-  and  dis- 
tlie  number  of  wheel-bhtdes  =  6;  the  blade 
radial  velocity  of  flow  at  outlet  =  2.877  f'-  per 
second;  the  velocity  of  flow  in  the  volnte  ami  discharge-pipe  —  j.JJi; 
ft.  per  second;  the  peripheral  speed  of  the  wheel  outlet-surface  = 
34.6276  ft.  per  second.  Ui-regarding  the  frictional  losses  in  the 
suction-  and  discharge-pipes  and  In  the  wheel-passages,  determine  the 
velocity  of  wliiil  at  outlet,  tli<-  blade-ti|)  angles  at  outlet,  the  deliverv 
in  cubic  feet  per  second,  the  speeil  in  revolutions  per  minute  .md  the 
aitii.d  lift,  the  efficiency  being  .759. 

Ans.  23.695  ft.  per  second  ;  ft  —  14'  44'  ;  73.13  cu.  ft.  ;  80.13  '■ 
19,34  ft- 

22.  A  centrifugal  pump,  with  an  actual  lift  of  10  ft.,  tlellvers  37. 85  cii. 
ft.  of  \vat<  r  per  secoml  at  a  speed  of  68  revolutions  per  minute.  The 
number  of  blades  -^  6;  the  blade  thickness  =:  |  in.;  the  wheel-dept!.  at 
outlet  =  9  ins.  ;  the  diameters  of  the  suction-  and  riischaigc-pipes  =  36 
ins.;  the  diameter  of  the  wheel  =  90  ins.  ;  fi  =  19"  7'  26.67"  ;  y  ~  9o''- 
Find  the  gain  of  liead  111  passing  from  the  wheel  into  the  volute  and  the 
frictional  loss  i/ir)  m  the  discharge-  and  suction-pipes  and  in  the  wheel- 
passages.     Also  find  the  eti'iciency. 

Alls.  2.193  ft-  :  '•';"  ft.;  -65. 

23.  In  the  ( entrifiig.il  pu.nps  for  two  torpeilo-boat  destroyers  the 
diameter  of  eye  =  7  iii.s.  ;  the  diameter  of  wheel  =:  20  ins.  ;  the  niiiiiber 
of  blades  =  6;  the  thickness  of  blades  =  ,\  in.,  the  width  of  the  wheel 
at  outlet  =  I,',  ins.;  tiie  actu.il  lift  =634  ins.;  cot  ft  =  5.167.  The 
pumps  are  driven  by  a  vertical  noii-i  ondensing  engine  with  a  45-in 
cylinder,  ;i  4-111.  stroke,  and  a  J-in.  pist(m-rod.     With  a  boiler-|)ressure 


5:(> 


FX.tMri.ES. 


'    1 


5 


i 


N, 


M 


of  220  ll)s.  per  -iquari"  inch  above  llic  atiiKisphcre  and  a  rut-off  at  \, 
llic  delivery  was  lound  to  be  1113  gallons  (U.  S.I  at  420  revolutions 
per  ii:imite.  Tlie  frictional  losses,  due  to  cue  upper  bend,  two  7-in. 
bends,  one  bad  check-valve,  one  gate-valve,  and  about  S  ft.  ot  7-1M. 
pipe,  were  respectively  estimated  at  .3//^,  .4//J,  /ij.  .1/'''.  .'uul  .41.S5  ft., 
//J  being  the  head  corresponding  to  the  velocity  of  discharge  (=  velocity 
of  flow  in  volute).  Find  (ii)  the  Tuechanic  al  efficiency;  and  also  find, 
on  the  oidmary  hypotl-.e.ses  and  assuming  y  =  yo",  {M  the  radial  velocity 
of  flow;  (<■)  the  loss  In  shock  on  entering  the  volute;  (,/)  the  hydraulic 
ctFiciency. 
A»s.  (a)  6.02  per  cent  ;  (/'i  4.014  ft.  per  sec. ;  (c)  61.7  ft.-lbs.  ;  (</)  .434. 

24.  Show  how  the  results  in  the  preceding  example  will  be  affected 
with  a  delivery  of  2000  U,  S.  gallons  at  an  assumed  speed  of  700  revo- 
lutions per  minute. 

A>is.  (<V  6.67  per  cent  :  (/-)  7.214  ft.  per  .sec. ;  (.  i  120  ft.-lbs.  ;  (</)  .^wj. 

25.  Determine  the  hypothetically  best  speeds  in  revolutions  per 
minute  for  the  pumps  in  Kxamples  23  and  24,  and  calculate  the  corre- 
sponding ma.\ii:Him  hydraulic  eiriclcncles. 

Alls.  In  K.\,  23  best  speed  =  292.7  rev.  per  min. 
••     ••     ::4     ••         "       =526       

26.  /.  centn.ugal  pump  delivers  20  cu.  ft.  of  water  per  second  at  a 
speed  of  225  revolutions  per  minute  ;  the  diameter  of  the  disch.irge-piiie 
is  20  ins.,  tile  diameter  of  tiie  wheel  is  36  ins.;  the  widtli  of  the  wheel  at 
outlet  is  5i  ins.;  the  number  <>l  bia<ies  =  6;  the  blade  thickness  =  i  in.; 
>' =  90';  cosec /i  =  1.4S4.  Find  the  liydraullc  eiriclency,  and  also  find 
the  tliameter  of  the  whirlpool-chamber  which  will  increase  tliis  ethci-m  y 

,,.//,      . 
Io35.!Sy' 

27.  A  centrifugal  pump  making  2295  revolutions  \)vr  minute  delivers 
23i  cu.  ft.  of  w.iter  per  second.  The  diatneier  of  the  dlschargc-pipo  = 
18  ins.,  of  the  wheel  —42  Ins.,  and  of  its  wliirli'o<jl-chamber  =--48  ins. 
The  width  of  the  wheel  at  outlet  =  3.452  ins.,  and  of  the  whirlpool-cham- 
ber at  its  outer  circumference  —  2.5  ins,  I'lie  tip  angle  /S  at  outlet  = 
cof  3.f>.     Assuming  the  ordinary  whirlpool  theory  and   disregartling 

hydraulic  re ances,  determine   iii\  the   radial    velocity  of  flow   ("v"); 

(/')  the  actu.il  velocity,  T'»,  with  which  the  water  leaves  the  wheel;  (f)  the 
/oit  in  entering  the  whirlpool-chamber;  './)  the  hydraulic  elhcienrv. 
There  are  six  blades  each  }  in.  thick. 

//«j.  {a)  9.35'i9  fl.  per  sec;  (*)  12.567  ft.  per  sec; 
(<■)  76.4142  ft.-ll)s.,         ((/i  .49. 


by  .I2J4. 


A>is.  .70S  — 


.»8  in^- 


INOFX. 


Abhot.  252.  2  =  3.  2?: 

Abrupt  changes   i.f   section,    loss  of 

head  due  to.  i()4 
Accumulators,  330 
Accumulator,  Brown's  steam,  144 

differential,  142 
Air  in  a  pipe,  1S3 
Air,  retarding  cflfcct  of,  224 
Applications   of    Hernouilli's    Theo- 
rem, T2 
Aqueducts,  circular,  242 

egK'^'haped,  244 

flowin,  240 

square,  243 
Arc  of  dischars;ein  oversh'^t  wheel, 

452 
Aspirator,  16 

Axial-flow  turbine.  41)0 

Hal.inc  inn  of  hoists,  345 

Harker's  mill,  375 

Harlow's  curve.  "3 

Harnes,  130 

Barometer,  water,  7 

Bazin.   230,   24?).   24S,  249,  25"'   2?*. 

257,  2?8,  jfiO,   2(16 
Bazin's  velocity  curve  and  formul.i, 

265,  266 
Bazin's  weirs,  qq 
Hear,  punching,  339 
Hcardmore,  247 
Heaufoy,  122 
Helgrand.  226 

Hclgrand's  sewer  formula,  246 
Hclidor,  386 
Hellmouth,  36 
Bends  in  pipe,  168 
Bends,  river,  26q 
Hernouilli's  Theorem,  8 

applications  of,  13 
Bidonc,  60,  384 


Rinding-press.  33? 
Hoileau,  2b8 

Boileau's    velocity    curve    ami    for- 
mula, 268 
Borda,  60 

Horda's  mouthpiece,  5S 
Horda's  turbine.  382 
Bordered  vane,  36S 
Hossut,  418 
Hourgngne    canal,   experiments  on, 

24q.  257 

Bovey's  tables  of  coclficieiits  if  dis- 
charge, 3q,  40 
Boyden's  hook  gauge.  2q8 
Hovden'sdiffusor.  4')2 
Brakes,  hydraulic,  353 
Hramah's  press.  33^) 
Branched     pipe     conncclmg      three 

reservoirs.  i')i 
Branch    main  of   uniform   di.imeter, 

18S 
Breadth  of   w.aer-uheels.  438 
Breast  wheels.  440 
Breast-wheel,  etiicicncy  of,  4^1 

losses  of  effect  in.  442 

mechanical  etTect  of,  442 

speed  of.  441 
Bres^c,  2qi.  2q2,  2q6.  3CK) 
Broad-crested  weir.  q4 
Brotherhood  hyilraulic   engine,  345. 
Brown's  steam-accumulator,  344 
Brumings,  247 
Bucket,  capacity  of,  458 

form  of,  43?,  458 
ii.'ckets,  number  of,  4if. 
Burdin's  wheel,  3S5 

Canal  lock,   time  of  emptying    and 

filling  a,  5q 
Capacity    o(    water-wheel    buckets, 

458 

577 


57'^ 


IMVX. 


•i 


5 


•« 


k4 


Capillary  phenoiTicnnn,  13" 
Capillary  lubes,  llow  in,  130 
Castel's     table     if     mouthpiece    co- 
efficients, (■() 
Centre  of  pressure.  \iv 
Centrifugal  force,  effect  of.  451 
Centrifuf;al  pump,  '(> 

analysis  of,  553 

efficiency  of.  555 

heiglit  of  suction  in,  554 

losses  due  t"  hydraulir  resistance 

in,  554 
values  of  <i,  /i,  ami   r  in,  55'' 
vortex  chamber  in,  5(15 
work  of,  }m,   555 
Ceiitrituijal  turbine.  303 
Chamtier.  whirlpool.  76,  505 
t  hannel-tlow  assumpti<nis,  22" 
Channel,    bottom    velocity    of    llow 

in  a,  266 
flow  between  liridijc  piers  in  a,  2<)(> 
flow  in  an  open,  221 
flow    through    contracted    portion 

of  a,  293 
form  of,  228 
maximum   velocity    of   t1,  iw   in    a, 

23(1,  25.S,  2(); 
mean  velociiy  of  (low  in  a.  2(iS 
mid-ileplh  velocity  of  flow  in  a,  2fi5 
of  ^reat   width  as  compared  with 

the  depth,  28S 
of  rectauKular  sei  tion   and   small 

slope.  287 
steady  flow  in  a,  221 
surface  velociiy  of  Mow  in  a,  265 
value  of  ((anil  fS  in  a.  249;  of  y  in 

a,  250;   of  II  in  a.  251 
variation   of  velocity  in  ^  section 

of  a,  257 
Channels,  cycloidal,  231) 

differential  equation  of  Ibnv  in.  275 

exam[)les  of,  228 

longitudinal  j)rofili'  of,  2S5 

of    constant     section,  steady    (I'lW 

in,  271 
of  varying  section,  flow  in.  271 
rectangular.  22q 
semi-circular.  238 
semi-elliptic,  239 
surface-slope  in,  227 
trapezoidal,  231 
with  change  of   section,  2()3 
with    constant    mean    velocity     of 

flow.  235 
Chezy's  formula.  163 
Cheirv's  experiments  on  Courparlet 

>  hannel,  247 


Circular  orifices.  81 
Cock  in  cylindrical  pipe,  i6q 
Cocks,  loss  of   head  due  to,  iftr) 
Coefficients,  hydraulii  .  29 
Coefficients  for  turbines.  ;io 
Coefficient  of  contraction,  34 

discharge,  38 

friction,  124 

resistance,  34 

velocity,  30 

viscosity,  269 
Cokcr.  130 

Combined-flow  turbines,  495 
Compressibility,  25 
Constants,  useful,  ivii 
("ontinuity,  27 
Contraction.  imi)crfect,  34 

incomplete,  35 

loss  of  head  due  to  abrupt.  165 
Coulomb.  122 
Coiirjiarlet  chaniitd,  experiments  on, 

247 
Critical  \  elocity.   I  2() 
Cunningham,  257 
Current-meters.  30^1 
Cylinders,  thickness  of,  337.  344 
Cylindric.il   body   in    pipe,   pressure 

otl.   406 
Cylindrical  mouthpiece,  63 

Danaldcs,  386 

D.ircy,  t26.  1315,  241),  2t.o,  303 

Darcy  gauge,  302 

D'Aubuisson,  126 

Uefontaine's  vclocitv-c  urve  formula, 

262 
Density,  2 
Diagrams  of  pipe  flow  expcrinu:its, 

1 40 
Didion.  403 
Differential  accumul.itr>r.  342 

equation  of  steaily  vaned  motion, 

275 
Diffusor,  Hoyden's,  492 
Divergent  mouthpiece,  66 
Downward  How  turbine,  490,  4(14 
Draft-tube,  theory  of,  529 
Drummi'rid  on  Miner's  Inth.  44 
Dubiat,  247,  258,  403 
Dupuit,  293 

KfJiciency    of     centrifugal      pumps, 

555 
Ktficiency    of     turbines,    conditions 

governing,  510 
remarks  on,  519 
cffecl  of  centrifugal    force  on,  508 


i\i>i:x. 


579 


r.iasticity  of  volume,  6 
Elbows,  loss  of  head  due  to,  \h~ 
Lllis,  177 

Energy,    losses     of     mcrgy    in    hy- 
draulic inachirics,  351 

lost  in  shock,  55 

of  jet  of  water,  09 

ot  water-fall.  7 

transmissif>n  of,  156 

of   fluid,  kinetic,  II;   pressure.  11, 
weight,  II 
Engine,  hydraulic,  347 

speed  of  steady  motion  in,  351 
EnlarRement  of  section,  loss  "t  liead 

due  to,  1(17 
Equations,  gener.il.  53 
Ei)uipotential   surf. ice,  20 
Equivalent  uniform  main,  l^(> 
Erosion  caused  by  watercourses,  227 

effect  o(,  226,  227 

table  of,  21"; 
L..>amples,  109,  2I(J,  32S,  335,40s,  •;3.). 

572 
rCxner    30S 

Expansion,  cubical,  tt 
Experimental  tank,  29 
I.\  telu  eiii,  247.  2^!i 

K.irnuT,  .(9.  Si 

Flamani,   144 

Float    adjustment    in    expe;  inir ntal 

tank,  41 
Floats,  suti  surface.  300 

surface,  3"o 

twin.  301 
Flow  from  vessel  in  motion,  2'' 

in  a  frictionless  pipe.  27 

in  aqucilucts,  240 

intluence  ot  pii)e's  inclination  ,ind 
posui<in  up"ii  the,  13S 

in  pipes    i  ;3 

in  pipe  of  uniform  section,  l.;3 

varying  diameter,  1.S4 
Fluid,  delinition  ol.  xiii 

friction.  121 

motion.  I 

pressure,  xiii 

rotation,  17 

whirling  ol,  19 
Foss,  143 

Fourneyrnii  s  turbine,  491 
Fournie,  142 
Francis,  80,  89,  301 
Freeman,  178 
Free  surf.ire.  20 
Friction,   coetlii  lent  of    124 

in  pipes,  surta'e,  125 


Friction,  laws  of  fluid.  123 
Frictionless  pipe,  flow   in,  27 
Froude,  131 

F'rou<le's    tabic    <d    frict.^jn.i!    resist- 
ances. 121 
Fteley,  S9 
Funk.  247 

Ganguillet  &    Kutter's   formula,  250 

(ias,  definition   of,  xiii 

Ganges,  experiments  on,  257,  278 

Gauckler,  253 

Garonne,  experiments  on,  2t)6 

{jauge,  Uarcy.  303 

Hook.  298 
Gauging,  methods  of.  297 

of  pipe-flow,  207 
Gaugings  un  the  Ganges,  27?;   Mis- 
sissippi, 253 
General  equations,  53 
Gerstner's  furmula,  421 
Grajdiical   representatiim    of    losses 

of  head,  170 
Grash(j(,  431 
Grassi,  0 

llagen.  139.  253 

Head.  27 

Ilele  .Shaw,  129 

Herschel.  20S 

Hoists,  hydraul.c  freight,  345 

Hook  gauge.  Hoyden's.  29.- 

Humphreys.  2?2,  253,  257 

Hurdy-gurdy,  4S5 

Hydraulic  coefficients,  29 

engine,  344;   analysis  <d.  347 

gradient.  13 

inltnsifier,  34-' 

jack,  33S 

mean  deptii.  222 

mean  raiiiu--.  135 

press,  335 

ram,  334 
Hydraulic  transmission,  i?fi 
Hydraulics,  definition  of,  I 
H  vdrodynaim. meter.  Perrcnlils.  308 
Hydrometric  pendulum.  308 
Hvdrostatics,      fundamental      prin- 
ciples of,  r.iv 

Ice,  weight  of,  3 

Impact.  359 
apparatus,  369 
roetficiem  of.  371 
III!  a  flat  vane,  3i;9 
on  a  curved  vane,  38.S 
on  a  hemispherical  vane.  307 
on  a  surface  of  revolution,  364 


cSo  INDEX. 


3 


9S. 


kij 


1. 

fi:          i 

Impact  on  a  v.me  with   hosiers,  30S   ] 
liiip.iLt  oil  .1  wheel .  378  i 

Imperfect  tontr.iction.  34 
Inclination,    iiilluencc  of   pipe's,  138 
Injector,  15 
Intensitier,  341 
Inversion  of   the    jet,  4S 
Inverted  sijihon,  li2 
Inward-llow  turbine,  4')0.  4')3 

Jack,  hydraulic,  33S 

lackson,  251 

Jet,  energy  of,  ficj 

inversion  of,  4S 

measurer,  37 

momentum  of,  69 

propeller.  373 
let    reaction    wheel,    375;  efficiency 

of,  376;  useful  effect  of,  376 
Jet  turbine,  4iXy 

Knibtis,  142 

Kutler,  I4'J,  230,  253 

Laminar  motion.  2 

I.ampe,  143 

l.esliros,  4S 

Level  surface,  20 

Levy, 143 

Lift,  balanced  ram,  345 
hydraulic  ram,  34') 

'.imit  turb'ne,  494 

Lines  of   force,  21) 

Liquid,  definition  of.  xiii 

Lock,  time  of   tilling  a,  50 

Longitudinal    profile   of  open  chan- 
nel, 2S5 

Loss  ot  energy  in  shock,  55 

Loss  of  he.id  due  to  abrupt  change 
of  secti(in,  164  ;  bends.  K.b  ; 
cocks,  H19  ;  contraction  of  sec 
tion.  I'll)  ;  elbows,  i()7  ;  enlarge- 
ment of  section,  1O7  :  orifice  in 
diaphr.igm,  166;  sluices,  169; 
valves,  i()<) 

Losses    of    head,     graphical    repre- 
sentation of,  170 

Losses  in  centrifugal  pumps,  55c) 
in  turbines.  531 

Magnus,  4S 

Main,  eijuivaleiu  uniform,  iH(j 
of  uniform  diameter,  branch.  18S 
with  several  branches.  201 

Manning.  230,  252 

M.iriolte,  403, 

.Metaceiitre,  xv 


Meters.  207 

inferential.  2oc} 

piston,  2o() 

rotary.  2o(; 

Schonheyder's,  20.S 

\'eniuri,  207 
Meyer,  261} 
Miner's  Inch,  44 
.Mississippi,    experiments     on,    253, 

267 
Mixed-flow  turbines.  4(j5 
Momentum  of  jet,  (jQ 
Morin,  403,  431 
Motion,  fluid,  I 

in  plane  lavers,  2 

in  stream-lines,  2 

laminar,  2 

permanent,  I 

steady,  i 
Motor  driven  by  w.itcr  flowini;  along 

a  pipe,  179 
Mouthpiece.  Borda's,  58 

convergent,  (>(> 

cylindrical,  63 

divergent.  66 

ringiio/zle.  61 

Navier's  hvpothesis.  203.  264 
.\'otch,  ?3  ' 

rectangular,  83 

triangular,     g2 
Nozzles,  174 

Ellis'  experiments  on,  177 

Freeman's  experiments  on,  TT"? 

( )pen  channels.  220 

Orifice  fed  by  two  reservoirs,  19= 

flow  through  an,  23 

in  a  diaphiagni,  loss  of   head  due 
t.,,  \uu 

in  a  thin  plate,  22 

in  vertical  pl.ine  surfaces,  78 

in  vessel  in  motion,  26 

with  a  sharp  edge,  22 
Orifices,  circular,  !-l 

large,  78 

rectangular,  7S 

senii  circular,  49 

triangular.  92 
Orleans  canal,  experiments  on,  247 
Outwaril  flow   turbine.  401 
Overshot-w  heel,  450 

arc  of  discharge  in.  452 

bucket  angle  of,  456 

division  angle  in.  456 

effect  of  centrifugal  force  in.  451  . 
impact  on,  469;  weight  on.  467 


isDt:x. 


58t 


Overshot-wheel,  450 

number  nf  buckets  in,  45''',  45 
pitch-iinKle  in,  457 
speed  of,  450 
useful  effect  of,  4(17 
weight  of  water  on,  452 


P.ickinR,  cup-leatlier,  336 

hemp.  330 
I'araboiic  path  of  jv:,  25 
I'draboloicia!  surface,  20 
I'aris  sewer  formula,  24^) 
I'astal's  press.  33b 
Path  of  iluid  parlirie  in  turbine,  486 
[V-lton  wheel,  4Sb 
I'eniiuli: 


hv<lroinetric,  308 


I'eriiianent   resiine,  1 


1 '  i  e  z 


•oclil's  hyilroilyii.imomete 


J08 


omettr.  i: 


Piobert,  403 


Fine 


connecting; 


thre 


reservoir 


branched,    njl,  20 


two    irser- 


ib; 


equivalent  unilorm 


1S6 


tluw  assumptions,  133 


flc 


.i^ranis,   144 


flow  in  frictionless,  27 
Williams'  cxixrinients  on  tb.w 

20b 
of  uniform  section,  How 
of  varying;  section,  184 
thickness  of,  158,  15O 


■33 


inclination   on, 


1' 


influence 


jf   pipe 


137 


Pumps     ce    irifugal,    £47:    analysis 
of,  553;   vortex  ch.iniber  'n,  565 


I'unc 


bear. 


Railiatinx  i  urrei.t,  72 
Ram,  hydraulic,  335 
Kayleigh,  Lord.  48 
Reaction,  373 

Reaction  wheel,  ctiicicncy  ol,  376 
Rectangular  oriuces,  78 
ROgime.  permanent.  I 
Reservoir  sluices,  q7 
Reservoirs,  branched  jiipe  connect- 
ing three,   icjl,  200 

oritice  fed  by  two,  i<)5 

pipe  connec'ing  two,  162 
Resistance  of  ships,  131 


of  motion  of  solids,  402 
Retarding  effect  of  air, 

nel  flow.  224 
Rtvy's  meter,  30O 


etc.,  in  c 


han- 


s.   I   Reynolds.  I2().  130,  139,  141 


f  velocity  in  transverse 


variation  o 

section  of    202 
Pipe-flow,   eff>JCt  of 

138 

Pitch  back  wheel,  472 
Pitot  tube,  302 
Plane  layers,  motion  in,  a 
I'oiseuille,  I2S,  131 
Puncelet,   4b,  41S 
I'oncelel  wheel,  424;  design  of,  433 


Rhine,  experiments  on,  247.  262, 
Ringnoiile,  61 
River-bends.  2(19 
Riveter,  portable,  338 
Rotation  i>f  fluids,  17 
Riihlmann.  2S5,  286.  293 

Sagebien   wheel>.  449 
Saone,  experiments  on.  257 
Schiele  turbine.  20s 
Schonheyder's  meter.  257.266 
Segner.  375 

Seme,  exiieriments  on.  257,  260 
Sharp-educ  orihces,  22 
Ships,  resistance  of,  131 
Shi'Ck.  energy  due  to,  55 
h)ss  of  energy  in.  55 


26b 


I'l 


ire  due  to,   160 


Simpson's  rule,  309 
Siphon.   iSl 


iverted.  182 


I'ractical   r.iefficients   in  centrifugal 

pumps,  569;  turbines,  519 
Piess.  Hramah's,  336 

Haling,  337 

hydraulic,  336 
I'rcssure,  centre  of,  xiv 

due  to  shoi  k,  l(>o 
I'reasure-head,  It 

on  cylindrical  boiiy  in  pipe.  40*] 
Pressure  on  thin  plate  in  l'il>e,  404 

of  fluids,  xiii 
■■'cny,  246,  24S,  259 
!>.-  peller,  jet,  373 


Slotte,  269 
Sluice  in   cv 


lindrical  pipe,  i6g 


in  recuingular  pipe,  i6g 
loss  of  head  due  to  a,  169 

Sluices,  437 
reservoir,  97 

Smith,  Hamilton.  Jiin..  87 

Snow,  weight  of,  3 

Sonnet,  260 

Specific  gravity,  xiii 

Spiral  flow  of  water,  75 

Standing  wave,  281 

Steady  How  in  channels  of  constant 
section,  221 


I  *. , 


<;8: 


iM)/:x. 


pipe  of    uni- 
,    dirterenual 


5 


N 


1 

B  VM 

ii 

■ 

L 

i 

Si.    ,. 

Sleaily    motion,  i  ;   in 
form  section,  133 
Sleaity    varied    motion 

e(|uatiun  of,  2u2 
Stearns,  S<) 

Storage  of  energy,  340 
Stream  line,  2 
Strickland,  4') 
St.  Venant.  24S 
Suction  tube,  tlieory  of,  sjq 
Siirfacetl')ats.   -,00 
Surfi.ce-friclion  in  pipes,  126 

slope  in  channels,  220 

tension.  41) 

velocity,  258-265 


Tables  of  back«ater   function,  290, 

2gl,  293 
bottom  velocities   2()i) 
Castel's  results.  6g 
coetfi<  ients  of  discharge,  39,  40 
coefficients   of  weir  discharge  by 

Fteley  iV  Stearns,  S<) 
density  of  water.  4 
discharge  through  Miner's  Inch, 4(1 
discharge    through    nozzles,    17"". 

178 
elasticity  of  volume  of  water,  6 
erosion  and  viscosity,  260 
expansion  of  volume  of  water,  6 
expansion  of  water,  4 
(rictional  losses  in  hose,  178 
mu>inuMn  velocities,  269 
f  and  »■  in  v  -:  <w/'i>.  153 
showing  best  relative  dimensions 

for  trape/'Mdal  section,  233 
slopes  and  mean  velocities,  227 
slopes   of  ttape/oi<lal   section,  231 
values  of  1  and  fi  in  Kazin's  form- 

uUc.  311  to  333 

f'r 

values   of  — rr=    for   centrifugal 

pumps,  55(1 
values  of  I'  in  Hazin's  formula,  350 

values  o(    '        for  turbines,  503 

values  of  m  und  h  in  (>=  ««{*  +  "'. 

316  I 

values  (if  H  tn  (tanguillet  tV   Km 

ter's  formula.  351 
vincijsity  of   water  .inil    nun  urv,   i 

value*  of  I  and  /■  in  channel  form-   ■ 
ul.T    c;.inpuiilet   .V    Kiitler.  323-   | 

32'-  I 


tube,  529 

pipes    and 
269 


Tables,  values  of  <    and   6  in   Man- 
ning's formula,   327 
Tachometer,  30S 
Tadini,  24'? 

Tank,  experiment, il.  20 
Tension,  surface.  4') 
Theory  of  suction  or  draft 

of  turbines,  497 
Thibault,  403 
Thickness  of    hydraulic 

cylinders,  337-344 
Thomson,  James,  77,  ()3, 
Thomsons  turbine,  565 
Throttle-valve,  loss  of  head  due  to, 

if,9 
Thrupp,  159 
Time   of     emptying     and     filling    a 

canal-lock,  ;o 
Torricelli's  theorem,  24 
Transmission  of  energy  by  hydrau- 
lic pressure,  136 
Trautwine,  S9 
Tri;  ngular  notch,  93 
Tub  wheel,  3S7 
Turbine.  a.xiai-How,  490,  494 

Hordu's,  382 

Hovden's.  49I 

centrifugal.  193 

combined.  495 

efficiency  ol.  501.  510.  mi) 

Fontaine's.  494 

Fourneyron, 491 

impulse  or   Girard,   482,  507,  513, 

517 
inward  How,  491,  493 
jet,  4i'<) 
jonval,  494 
limit,   494 

losses  ol   elfect  111,  531 
mixe<l-llow,  41)0,  495 
outward-tb'W    (ijl 
pi.rallel-tb'«  .  4  14 
practical   vaiui.s   o(   velocities   in, 

519 

radial-flow,  490 
reaction,  483,  516 
Schiele,  495 
Scotch,  375 
Segner.  37<) 
S»,iin'».  4')5 
l.ingenti.il.  393 
theory  of,  497 
Ihomson.  491.  493 
useful  work  of,  ;ol 
ventilated    4^1 
vortex,  491.  493 
WhileUw,  37S 


INHHX. 


5«3 


I 


Tutton.  146.  253,  2Sq,  2()I.  292 
■Pvvt'ddell  s  dirteri-ntKil  aicuimilator, 
34-: 

Umlcrshot-wheel,  4I() 
I'luk-rshi.t  wheel,  ictual  (iclivcry  in 
ft.•llJ^s.  of,  423 

depth  oT  crown  of,  431 

c-lhciency  of.  417,420;  I'oniel   •,4JS 

fiirni  of  course  of,  42i> 

in  a  str.uyht  race,  41-) 

losses  of  effect  with,  421 

modifications  toinLrease  elliciency 
of.  423 

number  of  buckets  in,  419 

I'liiicelel's,  424;  ethciency  of,  42S 

useful  work  of,  417,  420 

with  flat  vanes,  417 
Uiiilorm  main,  equivalent.  iSo 
Uiiwin.  4<J3 
Useful  constants,  xvii 


X,   and   !•  in  r  =  <>»/*(' 


Vallot,   143 
Values  of  . 

'53 
Valve,  loss  of  head  due  to  a    ii»> 
Vane,  best  form  of,  388 

cup.  3<i7 
Velocity,  bottom,  a6o.  266 

critical,  129 

curve  in  a  channel,  257 

formula-,  Hazin's,  2ii'i 

formuhe,  Moileau's.  268 

maximum,  2(10,  2(17 

mean,  258,  263 

mid  depth,  265 

of  whirl,  498 

rod,  3')i 

surface,  258,  265 

variation  of,  257 
Velocities     in     turbines,     practical 

values  of,  519 
Vena  contractu,  23 
Vt-nant,  St.,  248 
N'cntilated  buckets,  47a 
Venturi,  Wi  •rrnieter.   Id 
Vessels  in  motion,  orifiic  in.  j' 
Virtu.tl  fall,  Ij 

slope,  13 
Viscosities,  table  of,  369 


Viscosity,  264 

.Meyer's  formula  for,  2<)9 

Slotte's  fornuila  for,  2t)9 
Volute  of  centrifugal  pump,  558 
Vortex,  tircul.ir,  74 

compound,  70 

free,  74 

free-spiral,  75 

forced,  75 

motion,  74 

Water,  pressure  of,  6  1 

weight  of,  2 
W.itcr-barometer,  7 
Water-meter,  207 
Water  pressure  engine,  347 
Water-wheels,  class  ticatioii  of  ver- 
tical, 416 
Wave  propagation,  velocity  of,  161 
Weight  of  fresh  '.vatcr,  3 

of  ue.  3 

of  s.ilt  water,  3 
Weir,  83 

Kazin's  flow-over,  99 

Heam,   104,    107 

broad-crested,  94,  10(1 

drowned,  88,  lob 

inclined,  89 

rectangular      with    end     contrar 
lions,  86  ;  without  end  contrac- 
tions, 85 

sharp-crested,  99,  107 

submerged,  .^8 
Weisbach,  36,  60,  166 
Weser,  experiments  on,  247 
Wheel,  breast,  440 

hurdy-gurdy.  485 

in  straight  race.  41S 

jet  reaction,  37s 

overshot,  450 

I'eltnn,  48(1 

pitch  back,  473 

I'onceirt,  424 

Sagcbicn,  449 

undershot.  41O 
Whirling  fluids,  19 
Whirlpool  I  hamber,  7& 
Whirl,  veloi  ity  of,  510 
Wliilelaw,  375 
WillianiF,  200 
Woltmann,  247 


jawijifh 


"i^.: 


r 

i 

1 
} 

SHORT-TITLE    CATALOGUE 


mUJCATlLLNft 


juji^'  w  1  M-:  ^'  \-  SONS, 

New    York. 
Lo.\r...\:    CIIAI'MAN     A     il\!.l..    Limukd. 

AI!i;.\Ni.i;i>  UNDER  Sl'IiJECTS. 


Iipscriplivc  .•irvularM  si'iil  otj  npplication. 

It..i.ks  ii>nrk>-cl  wiili  an  iitN-iisk  arc  si. 1.1  at  iirf  pricps  only. 

All  books  are  bound  in  elotli  unlehs  otli<'rwi»>-  istati-U. 

AdRICl  I  TlPn. 

Arinsby's  Mnnuiil  nf  CaItU- Feeding 12mo,  |1  75 

Dowiiitig's  Fruit  1111(1  Fnilt  Trofs 8v(i.  5  00 

GR)t.-iif<!fs  Tin-  PiliKiplts  of  Modern  Dairy  Pructke.     (Woll.l 

I'Jin.,  2  00 

Kimp's  I.iiriiiv^cupr  (innloiiiiig Iv-'iiia,  2  50 

y'lyimril's  Laiul'^cnpi-  (iiiplcniiip 12nio,  1  50 

^     el's  Trnitise  on  tlio  Disi'iists  of  llic  Dog Hvo,  3  50 

8io(kliii.|gc'«  H"ck- Hiid  Soils Hvo,  2  50 

■yVoir^  Iliimlbook  for  Fuinin-iiiMl  I>:iirvmon 12mo,  1  50 

AKCtllllCIl  kl.. 

Bergs  Buildings  nil. 1  .simcuirfs  of  AiiktIiuii  UiiilroiKis 4to,  '>  00 

Birkniiif'ii  Amerinin  TlunlreH-I'liinningund  Const  ruction. Bvo,  a  00 

"        Aiiliitictiiriil  Iron  find  Stetl 8vo,  3  50 

"        Compound  Hivricd  (iiidors 8vo,  2  00 

"        Skek'tun  Cooslructioii  in  nuililingH 8vo,  3  OO 

••         Planning  and  ('..iisiiuction  of  llii;li  "t'i<'"'  HulMIngri. 

8yo,  3  50 

Brigp«    Modrrti  .\mi    Scbo(d  Building fvo,  4  00 

C»iptnl«i'i  lltatiug  uml  Vi'ntlluliui:  '>f  Buildiugs fivo,  8  00 

1 


1 1 " 


h::^ 


Kic:t!iL''>  ArcliitiM  :iu;il  F.iuin  Clin;; Svo.  f'^  .",0 

The  Kiri-prooliiig  of  Steel  Huildings Svo.  2  50 

(JiiliiirilV  Sanitury  House  Iii-|i<-rlioti Ifiiiio.  1  00 

T!ir;itif  Fir.s  iiiiil  !';mic> IJiiio.  1   "jO 

Iliiilii'il-  AiiH'iicHii  lIoiiM- Caipfiiter ><vo.  ';  00 

Holly's  (  .irpcnU'r  iiinl  .Toiiier ISiiio.  7." 

Kiililcrs  Aicliili'it  ami  IJuiMci's  I'oiUit  hook. .  .Iflmo,  morocco,  4  Oi) 

MciriUs  Stiiiii's  for  UuililiiiLT  an. I  Decoration Svo,  ."5  00 

Menu  kloii's  Stair  l'.iiilcIiiii:-Wooil.  lion,  and  Sloiie 4to.  4  00 

Wail's  Kiiiriiii'crinir  anil  Aicliiliciui:il  .liirispiuilence Hvo.  6  i>0 

Sheep.  6  50 
'■        Law   of  operation-  Pnliniinaiy  to  Constriitlioii  in   Kn- 

jrineerihi;  ami  Ariliiltctiiri' f'vo,  ."i  00 

Sheep,  5  50 
Woreesler's  Small   Hospital— Ksialili-hnient  ami  Maintenance. 
incluiiiiii:   Atl>inM)n"s   hujigistioiis    for    Hospital    Arclii 

teclure rjnio, 

•  Worn,  s  Coliiniliian  K.\|i..siii,.n  of  isy;; Laru'e  4lo, 

\kr  \n  ,  NWS,  ntc. 

Uernailoii'a  Siiiokele>~    l"n\Mli  r   K'mo, 

•  HnilT's  (  Milnain-e  ami  <;iiiin(  ry f^vo, 

C'lmse's  Screw  Propellent  ^tvo, 

C'loiiUliile'-  (luiinery  for  Non  riJin.  <  Mlicers 32ino,  iiioriK'co, 

•  I»>.vi-'s  Treat!- Military  f,a« ><vo. 

Sheep. 

•  Klenienta  of  Ijiw fvo. 

I(e  lira«k'- Cavnlry  <>iiipo-i  Duiii  »      ,('arr  i   ,  .  .liSino,  nioroieo. 
Diet/'s  Soulier's  Fii>t  Aiil  Itlmo,  morocco. 

*I)reili;e's  .Modern  Kren<  h  Ai  iiliii'.\ . .   .Latuc  4to,  half  nioroi'co. 

•  "  Uecord    of   the   'rraiisportatioh    Kxliihits    Hiiildiiiir, 

Wtirhl'.sfiiliniiliiaii  Kxposilionof  lS9H.,-4t<i,  half  morocco, 
Duniml's  HeKlHlani-e  ami  I'ropulsioii  of  Sliips Xvo, 

•  Kiiliegcr'H    Kiclil   Kortiticulion.    i'lcliiding   Mililury    Uridires, 

|)t>niolition-.  Kn<  atiipiiKiiitK  and  ('ninniiinicalions. 

Liiricc  I'Jnio,  y  00 

•  l»\ei  -  1  lu'i'i  Arliiii  ry .12mo,  :t  Oil 

•Holls  Naval  Taclic!. "vo,  1  SO 

•  luKalln'ii  Ballistic  Tallies i  50 


1 

•25 

1 

01) 

■» 

50 

0 

m 

3  1)0 

3 

00 

1* 

(M) 

1 

50 

2 

50 

•i 

00 

1 

'.'5 

15 

M 

>> 

(10 

5 

00 

ln-;iils'>  lI:iiullM...U  ,.f  Pro!i!fiii.*iii  Dirci-t   Fin- ^V". 

Mtiliiiir>.  IVnmiiRM.t  Foililicsiiinus.  iMcivur.  )>vo.  iiiilf  inonKC", 

M :,„ii;il  lor  (■o.iit>  MHiliul I'i'"".  """"'L'". 

»  M.nui  ■>  Attn.  U  ..f  Fonili.  .1  IM:.. .  s Vhno. 

*  ••  KkiUfiiN  nf  til.'  Art  <^f  \V;ii »v.., 

McUiilfc's  <  >i.liiaiiiv  >xiu\  Ciliim-iy Vinu,.  witli  Alias, 

Muiiav  -  Ihfmitiy  I>iHl  H.-L'tihilions  !i<iai)HMl  t..  tlu'  Si.riiiglicl.l 

UillL..  Caliber  .45 32iiio,  pai-.T, 

•  I'liilps'^  I'lactical  Mariin' Survi-viiii; ^vo, 

I'.i'AfU;.  Aiiiiy  OlliciTS  KxaiiiiinT r-'iiii>, 

Sl,:u  IH'V  StilwiMinj;  Annies ••'■-'"'.■.   i.i..ro.  c... 

WIii-iUts  Si.-c  < )|..TatiMtis ^^'"' 

WiiiUiK'p's  Aluiils-'mciil  cf  Mililan  Law Vinn>. 

\\ llnill's  Xntf>  on  Military  llysiifiif I'ii"". 

Y<.iiiin"s  Siiiipl.'  Kl.-inoiits  of  Navigation    lOmo.  morocoi. 


14  00 
7  .-.() 

1  Ol> 

2  dO 

4  IH) 

5  00 

10 
4  00 

1  r.o 

2  00 
■i  50 
1   50 


.  .Nvo, 
.I'iino, 
.I'Jnio, 


2  00 
tirst  fililioii 1  OO 

ASSAVIN(i. 

Fl.'tclu'r's  Quant.  Asstiyinji  with  lla-  lUuwi.ii«.>..lt5ino,  morocco, 

Fiinnan'-  I'raiti.  al  Assaying '*""• 

Kuiilii.nit  s ( )ie  Dri'ssin.i: ^'"• 

Miller's  Matiual  of  A.osayiiiL'. '-'»"• 

ODriseolisTi  eat  mint  of  liohl  Ores   8vo, 

Uieketts  and  Millti  -  Notes  on  Assaying: f^vo, 

■riiiirston's  Alli>ys,  Hius-es,  ami  Uionzes 

Wilson"*  I'yaliiile   I'iciees.-es   

The  I'lilorinutioii  1'i«<'<'<s. . 

t'laiij'H  Aziinulh   *^" 

Doolitlk-s  I'lartieal  Axlronoiny 8^'"- 

(loie"»  Kleiiifiitsof  C5e.«li'sy *'"• 

Uiiyforil'n  'I'ext-lHMiU  of  Ueoilelie  Aslrononiy .  ^vo. 

♦  Miehie  ami  Harlow  V  Piaeiieal  Aslrononiy >*^'". 

•  Wliite.-,  Tlieon-iieul  and  I).-erii.'ive  Aslromiiny 13ino, 

Hilldwlii*  Orelii.lsof  New  Kmilan  1  -mall  >*vo, 

'riioim''«  Strmtnral  Uotniiy 

NVi'sleriinders  general  Hottmy.     (Schneide' 


Kiinu, 
■svo, 


1  50 
3  00 
1  50 

1  00 
••  00 

3  00 

2  (W 
1   .50 

1  50 

:!  .50 

4  00 

2  50 
n  00 
:i  00 

2  00 

1  50 

2  3.'> 
2  00 


fe:| 


nWIfXir.S,   ROOFS,    F.tc 

(Sei  also  KNiiiNEEliiNii.  p.  7.) 

Boiler's  llidiway  BrML't-s 8vo.  |2  00 

•     ••       Tin-  Tlrmi.-  HiviT  Briilgc  4t(>.  r"!'*-!".  ^  •"> 

Biiir-  Siri'ss<-s  in  liridL'is. f^vo,  3  50 

Crehoii's  Mwluiiiifs  ..f  tin-  (Jinler 8vo,  5  00 

Dii  IJnU's  Slrc-ses  in  FniiKii  Struiti  ■  •« Small  4lo,  10  00 

Foster'.- Wooiliti  Tresll.- IJiidgfs  4tu.  5  iM) 

Grccuf'.s  Airlic*  in  AVuoil,  itr f'vo,  2  TA 

"         Hii.lKc  Tru-^ses 8v»,  2  r,0 

U.Mif  Tru<-sc8 8vo,  1  25 

Howe's  Trr.itise  on  Anhi-" 8to,  4  00 

Jolin-nn's  :M.i(liMn  Fnnicd  Slrudiirps Small  4tii,  10  00 

lltrrini:in    iV    .I;ni)li\ '.-i     T.\l  Imok     of    Roofs     and    Rridgps. 

I'ait  I.,  Stnssis 8vo,  2  50 

MerrinKin    iV:    .Tacoliy's     Ti'MImm)!;    of    Uoofs    and      bridges. 

T'ait  II..  (iraiiliii'  Stalies  8vo,  2  50 

Morrinian    iS;    Jai-oliy'-     Tixt-liook    of    U,,of>    and      HiidL'i'S. 

I'ait  III.,  Hiidirc  Design -^vo,  2  50 

Merriniaii    i\:    .lacoliys     Te.\l-'>ooU    of     Uitofs     and     Hridj^es. 
I'art  IV..  ('onliniious.  Draw,  Cimlilevcr,  Suspension,  mid 

A I  ( li.  d  liiid-e- 8vo,  2  .50 

•Mori-on's  Tlie  M.  nipliis  Hrldf,'e 01«lon>r  4to,  10  00 

■Wiuhlell'H  De  ronlibu.s  (u  Pocket-book  for  Bridge  Engineers). 

lOmo,  morocco,  3  00 

Specitieiilions  for  Sleel  Uiidges 12mi>,  1   25 

Wood's  t'onstmclion  of  Hridj'es  and  UiM)fs f*vo,  2  <X) 

Wriglit'a  Uewguiug  of  Dr.iw  Spans.      I'arls  I.  and  Il..S<vo,  each  J     ) 

••               "          "      ■•           '■          C'ompUti 8vo.  3.50 

CHn.MISTRN      lilOl.OCiV     PHARMACY     SAVI T AkV  SCIliNCi;. 

Adriimce's  Laborutory  CaUululiong r.'mo,  1  25 

Allen's  'raltles  for  Iroi>  Analy.sis 8vo,  3  00 

Austen's  Notes  for  ("Inmical  Stiulwitii 12ino,  1  50 

Bolton's  Student's  Ouide  in  QnantitnliTC  Ansly^in 8vo,  1  50 

CluBscu's  Analysis  by  KIcetriilyfls.   (Uerrickuud  Boliwood.l.Svo,  3  00 

4 


#! 


Colin "s  Iiidicators  nnil  Test- papers I'^iiio 

Ciiift'^'s  (^iialiiMtivi'  An.'ilysis.     CSclmclTiT.  i l'.3iii(i. 

Diivcnporl's  Stati>ti(al  Metliixl-.  with  Special  licfercnce  to  Hio- 

l(.)Lrical  Vuiiatioiis 1 -iim.  iiiori)(<u, 

DruL-hsel's  Clieinical  lieaclidiis.    (Mi'iriU.) I'-iin... 

Kniniaiiii's  IiitrtMlurlioii  tu  Clit'miial  Prepaialion.-.      Diinlap.  i 

I'iinii, 

Fie-ciiiu>-"s  Quail  I  it  at  i  VI-  f'ln-riii(  :il  Analysis.    (Allen,  i 8vc>. 

yualitalivf  •  -.I(.hiiM)n. "^vo, 

(Well-)         Trans. 

lOlli  GiTiii.in  E'litiou ^vo, 

FmrtesVs  Water  and  I'liMic  Hfaltli Vi\u<\ 


1  .■,() 

1  2,1 

I   -J.-, 

1  2« 
6  00 
3  00 


AVatcr  Filtration  Works 1  Jtuo, 

Gill's  (iasaml  Fuel  Analysis I'-inio, 

Goodrich's  Kciitioinic  I»ispi>sal  of  Tovvn.s'  Kcfiisc Druiy  8vo, 

Hiinniarslcirs  l'liysii.In;ric'al  Clumisiry.    iMandel.) Svo, 

Holm  -  Prii.ciplis  of  Mallunialica!  (In'ini^lry.     >ror};:iii)  I'inio, 

Hopkins' OilChiinisl'^  Ifand-booli -v., 

Ladd's  (J\iaiititativc  flicnucal  Analysis I'-^nu,. 

I.auilamr's  Spectrum  Analysis.     jTin;;li'.  i f^vo, 

I.iilis  Electrolysis  and  Elcctrosyntliesis  of  Organic  t  oiiiptmnds. 

•  F.orcnz. ) 12nio, 

Malidel's  iiio-i  lieinical  I.alioial'Ty 12mo, 

Afasoii's  Water-supply Svo, 

Kxandmitii'ii  of  Watt  r I'imo, 

Meyer's  IJadides  in  Ciilion  Coinpouiuls.  (Tingle. ) I'Jnio, 

Mi.Mer's  Klenienlary  Tixt-hoid;  of  C'li.-mistry ISnio, 

Morgan's  Tlie  Theory  of  Solutions  and  its  Results Timo, 

Elements  nf  Physical  Chemist  r_\ I2ino, 

NiihoN's  Water— upply  iCliemical  and  Sanitary 8vo, 

<  ilirine's  Lalioralory  Guide  to  Chemical  Analysis yvo. 

Pinmr's  Organic  CluMuistry.     (Au.sten.) 12mo, 

PoiOe's  Calorific  I'o'.verof  Fuels 8vo, 

Richard.s's  Cost  of  Liviugas  Moililied  \>y  Sanitary  Science..  12ino. 

"        and  Woodman's  Air.  Water,  and  Fooil 8vo, 

Uicketis  and   Uussell's   Notes  on    Inorganic  Chendsitiy  (Non- 
metallic  i ( tlilimg  wvo.  nioriM-co, 

Hideal's  Sewage  and  the  Bacterial  I'urification  of  Sewage..  8vo, 

15'iddiinan's  Incompaliliilitics  in  Prescriplinns 8vo, 

Sihimpfs  Volumetric  Analysis 12mo, 

SiMMicer'»  Sugar  Miniufacturer's  Handlxiok.   ...lOiiio,  morocco, 
'•  Hniidliook    for    Clietni!>is    of    ll<\.t    Sugar    Ilousen. 

Irtmo,  moroc<o, 

Sto<kljriilge's  Itoeks  and  SoiU  fvo, 

•Tillman's  Descriptive  General  Chemistry 8vo, 


r,  00 

1  50 

2  50 
1  25 

3  30 

4  00 
1  50 
■!  00 
1  00 
3  00 


00 
50 
00 
25 
1  00 


1  50 

1  (10 

2  (K) 
2  50 
2  00 
1  50 
;i  00 

1  110 

2  00 

75 

3  50 
2  00 
2  50 

2  IN) 

3  00 

2  50 

3  00 


Vail  Di'VoiitcrV  rii\>ii;il('hcinistr_v  fur  IJc^'i'.iiurs.     '  BultWdOii.) 

12ni,i,  |;l  50 

\\\'\  .s  liioiiiiuiii- Qu;iiitul'n.>  Analysis 12rn(>,  1  50 

Lalmratniy    Giiidf    in   liiialiialive   Clicmital    Aii.il,\  >is. 

Svo,  1  50 

Wliii'pic'-  Mi(M>r<iiiy  of  MiiiiUiiiir-watiT f*v(),  :i  50 

WiiTliiimmrs  clicniiral  [.i-ciuic  Nctcs liiii".  :(  00 

SiiL'ar  AMaly-i> ^^niiiU  f<v<).  2  50 

Wulliug's  Iiiorjiaiiic  I'liar.  ami  Mcil.  (  liiinistry r.iiio,  2  00 


•^^ 


t^ 


I)RA\MN<i. 

■*  liartlett's  Mecliaiiical  DiawiiiL' S^vo, 

Hill's  Shaiii's  and  Sliadmvs  ami  Pcrspfclivc fvo, 

Ma((  niil's  Dcsriiiiiivi-  (k'DiiR'try **vi>, 

Kiiu-maliis f^vo, 

'Mi-chaiiiial   DiHwiiii: i^vo, 

Malians  Imltistiial  DiawiiiL'.     ('riinnipsdii.  i 2  Vols.,  Svo, 

liicds  Topiiirrapliical  DrawitiL'.     (H.  A.i -Ito, 

IJiid's  A  ('(iiir^r  in  Miclianiial  Dniwinir '^vn. 

.Mcchaniral   Diawintr  and  EliiniMitaiy  Madiint-   Di'>ijrn. 

Svo, 

Snd Ill's  '''opd'^rapliiral  Diaw  iui.'.     (Min'millan.  i Svo, 

AVaiicn's  Dfsitiplivc  (tcMiiic-liy 2  vids.,  Svo, 

Itral'linir  Insiiuimnts 12ino. 

Frcr  liand  Diawinj; IJmm.. 

I. hilar  Pcrspcclivf 12iu<i, 

Mai'liinc  ('oiistniclion   2  vols,.  Svo, 

Plane  I'rolilcms 12ino. 

Primary  (u'omctry 12nio, 

Prnlili'Mis  and  Tliuortins Svo, 

"         Projciliori  DiawinL' 12mo, 

"         Shades  and  Shadows Svo, 

Slereoloniy— Sotu'-cullinj; 8vo, 

^^l\elpley■s  Letter  Knjiravini: 12nio, 

Wilson's  Free-hand  IVr>pfelive Svo, 


;5  00 

•) 

00 

:; 

00 

5 

00 

4 

00 

:{ 

50 

5 

(to 

o 

00 

3  00 

o 

50 

:i 

50 

1 

25 

1 

00 

1 

00 

■*■ 

50 

1 

25 

75 

o 

50 

1 

50 

;i 

00 

2 

50 

2 

00 

o 

50 

HIIClklcilA.    M\»JNfTI5M.    \M>Pin>lC5. 

Antlionv  and  Itnuk' lis  Text  l.^ok  .if  Pli\-ic  .       .Ma-ii    I   Small 

Sivn,  3  00 

A'llhoiiv's  Tlieorv  of  Kleelrienl  Meiisurenieiits 12nio,  1  00 

Ilaiker's  Deep  sea  Soundings Svo,  2  00 

Benjamin's  VoliaieCell   Hvo,  3  (K) 

History  of  Kleelrieity Svo,  3  00 

0 


Cla-senv  Analysis  l.y  Electrolysis,    (llfviick  and  Boliwood  i  8vo,     $3  00 
Cnhoiv  una  Snuicr-.s  KxprriuuMits  rvill.  a  Now  Polanzmj;  Phct..-      ^ 

(  lin.iKiirrapli 

Diw-nns  Klciiric   Ki.ilwavs  uii.l   Tramways.     Mnall,  4lo.  liall 

iiiorocco,     r-i  ■)*• 

1(1  Klcclrii"    Tiaclioii  rockct-lxi  'k.      16mo. 

inoincco, 


'  KiiiiiiiLiTiiii; "'  »u<' 


(K) 


Drc'lirt-'s  E'.'-''tric-  Illuiiiiiiali"iis. 


.2  vols..  4lo,  half  lUdioccn.     25  OH 


Vol.11 -l.o.       7.W 

GillMifs  l)c  maf,'iielr.     (Moiulay.) ^^""^       '•'" 

Holmaiis  Precision  of  McaMircincMil- "^''^       -  _^_ 

T.Wo,u-ii.irror..scale  M.tl.o.i Large  Svo, 

Le  Clmu-lier-s  HiL'l. 'iViniH-ralures      .Bur-e-s) limo,       .5  00 

I,M,V  Kl.'Ctrolvsisaiul  ElecUosyiilhfsis  of  Organic  Comi.ouii.ls 

.     . ,  ./)  12iiio 

Lyons's  Eloc'tro.nagn.-tic  Phenomcim  a.i.l  ti.e  Deviations  of  ihf 
ronil>a.ss 


1  00 
f^vo.        0  IM» 


•Miehies  Wave  .Motion  KcOalinf:  to  Sound  aii.l  I.ipbt «vo,  4  00 

jionjans  The  Theory  of  S..liilioMs  anil  its  Itesults V-imo.  1^  •_»> 

Nia.nl.l  V  Kleeirie  IJatteries      t  Fisiil.a.k. ) ■  l^'"".  -^  •'" 

•Parshall  &  Hobarl  Kleeirie  tJeneraior.s.     Small  4to,  luilt  mor.,  10  (K 

Pratt  and  Alden's  Sireetrailway  Uoad-hed- ^vo,  2  IH) 

Uea.'tms  Sl..ani  and  ICleilric  Lrn'onioUves l-'i'io.  I  00 

Th.i.-I..ns  Stationary  Slean.  En,Lnne,«  for  Kleetrie  U^hUn'A  Pur- 

8vo,  2  50 

.^•n  ''"**■':;■;•• '.".'. hvo,  i  so 

•Tillman  s  Heat 

Torv  .V  Pitcher's  Laboralu,  v  I'i.VM,  s bmall  8vo,  2  00 


(•,v,,,_Mi,  iMNK  .\i.-S\Mr.\tiv.  Err. 
<i,e  .,l.o    Bhiihm.>,    p.   4;   Hvi.uAn.us,    ^   9;   .Maik.uai,.;  - 

..INKK.UIN...  1<.    11  ■.    MkC.IANKS    AM)    MaCHINKIIY,  p.   U'  . 
ESOlNKS    ANO    P.oll.Kl!S,   ]>.   14.) 

Baker's  Mas<i|iry  Constrm  tioii ■•  j^^"' 

Siirveyins;  Instrunienls I'-ino, 

Blaek's  U    S.  Pul.lie  Works l»t'i"".f  ■'I"- 

Btooks's  Sireetrailway  Location 10'"".  nioroc.-, 

Bntts's  Civil  En,i:i..eers'  Field  Book ICmo.  tnorocco, 

J'.yim's  lliirhwav  fonslruelion ■^'^'"• 

Inspecliou  of  Materials  and  WoikniHn.sliip lOmo. 

Cirpenler's  Experimental  Knu'lneerinc  8vo, 

Church's  Mechanics  of  EnirineerinK-Solids  and  Fluids.  ...8vo. 


W    F.N. 

Stkam 

6  00 

3  00 

5  00 

1  .-lO 

2  50 

T)  00 

;!  oil 

6  00 

6  00 

ii;iJ 


/-I 


'Ifk 


\ 


hi 


C'liurcli's  Xoti'S  ami  Exiiiiiplcs  ill  Meciiauic.-. -^v...  |J  00 

Cniuiliill's  K.irtliwnik   Ttilili-s svo  j   -,0 

Tlie  Transition  Ciirvf lOmo,  morocco,  1  r.O 

Diivij's  Elevation  auil  Stadia  Tables Small  8vo,  1  uo 

Dredge's     Pi'im.     Hailrnad     ('(instniotioii,     ett'.        Larirf     4to, 

half  iiiurocco,  ^\ii.  paper,  5  no 

*  Drinker-  ■runiirlliMj; .(t...  iialf  miin.ceu,  •,>.■•)  no 

Eissler's  Kxplosives— Nilroglyeerine  niid  Duitdiiite 8vo.  4  0(» 

Frizeil's  Water  I'ower svo,  .")  00 

Folwell's  Sewerage gvo.  3  00 

Water-supply  Engineering Svo,  4  iiO 

Fowler's  C)fTer-dain  I'roccss  for  Piers n\„'.  .>  :,ii 

Fuerles's  Wilier  Filtration  W^)rks lOinn,  *;  V> 

Gerhard's  SanituiT  House  Iiispeetiou IJimi.  1  01) 

Godwin's  Uaiiroad  Engineer's  Fieldtiook Iflnic.  niurocen.  '.>  :.i.i 

Goodrieh's  Eco :Ai-  Disposal  of  Towns'  llefuse Demy  Svo,  3  ."io 

Gore's  Ehineiu     ..Geodesy ".Svo,  v'  ."lO 

Huzlehiirsf s  Toweis  and  Tanks  forCities  aud  Towns Hvo,  -'  ."lO 

Howard's  Tiansilion  Curie  Field-book IGnio,  uiorocco,  1   -lO 

Howe's  HetaiiiiiiLr  Walls  (New  Edition.) r.>mo.  1  V> 

Hudson's  E.\eavatiou  Tables.     Vol.  II svo,  1   00 

HultDii's  Meehaiiie;d  Engineering  of  Power  Plants 8vo,  ."i  no 

Ileal  and  Heat  Engines f^vo  '>  00 

Johnson's  Materials  of  (..'oustrueiion .sjvo,  ti  00 

Theory  and  Practice  of  Surveying Sinali  8vo.  1  00 

K'  tit's  Mechanical  Engineer's  Pocket-book ICino,  morocco,  .")  no 

Kii-rsteds  .Sewage  Disposal I'^nio  1  •,>.■> 

Jlahan's  Civil  Engineering.      (Wood.) Svn.  :>{>{) 

Meniman  and  Mrook's  Handbook  for  Surveyors.  .  .  .lOinn.  nior..  ','  oO 

Mil  1  iiiian'--  Precis*  Surveying  aud  Geodesy Svo,  li  00 

Saiiilaiy  Engineering yyo,  2  00 

Nagk's  Manual  for  Uaiiroad  Engineers lOmo,  uiorocco.  "  00 

Ogden's  Sewer  Mcsign io,„„'  .j  ^o 

Pations  Civil  Engineering Svo,  half  morocco,  7  50 

l''"""''''"''>"s 8vo.  5  00 

Pliilbiick's  l-"ield  Manii.-d  for  Engineers Ifimo.  morocco,  :',  00 

Prall  and  .Vldi'ii's  Street-railway  Koad-beds yvo,  ^  (lO 

IJockwell's  Uoads  and  Pavements  in   I'laiKc.    IJnio,  I  0." 

Schuyler's  Reservoirs  for  Irrigation Large  Svn,  tt  oO 

Settiles's  Field  Engineering i6mo,  mnrocc.  y  nO 

Railroad  Spiral lUnio,  iiinrocco,  1   :,u 

Sielii'i'  and  Higgin'-  .M.idern  Stone  Culling  and  Masonry. .   sivo,  1   .-,0 

Smart's  Engiiipcrin^'  Laboratory  Practice ".  .13ino,  2  .%0 

Smith's  Wire  Manufacture  and  Cses .Snnill  4to,  ;)  ("»0 

Spaitiiiigs  Kn!id>  toiii  riivcments , 12m.i  2  00 

3 


■Spaldiiijj's  Hvihaiilif  Cemenl IJm.i. 

T!ivli>r'>  Pri.-moiilul  Foriuul-'i  !iiui  Kutiliwuik 8vi>, 

Tliurstous  Miiteriuls  of  Couslnictiun   <*Vf. 

Tiilson  s  Mnei  I'livoiiieuts  unci  Pavini;  JI:ileii..'s jvo, 

*  'rruutwiiiuV  C'i'.  i'l  Kiii:iiieei\  I'iK.Ui;t-l)o(.k . .  .  .Itimu,  morocco, 

*  "  (■r(l^^■»^■cli'lll Sluct, 

»  "  Ex<iivati(iiis  and  Kiiibaiikmcnts ^vn, 

*  •■  L:iyiiig  Out  Cuivcs T-iiiio,  iiuirnicD, 

Turiieauiv  umi  Uusscli  s  Public  WalerM)|)[i;ii> 8vo, 

\V;uliicn'>  I>i'  Poiililms  (A  Pockfl-bn :.k  tor  liri.l^M'   ICiigincfisi 

liimu,  iiiDrocco, 

Wiiil-  KiiL'tiiceriiiK  aiul  Arrliitectmal  .lurisprudinri' .''vo. 

Sliccii, 

"      Law  lif  Field  Opcrali       in  Liigiiiccrins:,  etc 8vii. 

Sheep, 

W'.irrcn's  Stcicdtotiiy— Stoiiecutliiiu' 8vi). 

Wcbl)'sEu.!;inet'iing Instruments.  NewEdilimi.  Ifiiiui,  iiKirncct), 

Uailioad  Const  ruction ^\o, 

AVi  ;;m:nurs  t'oiistruction  of  Masonry  Dams Ito. 

Wclliiiirion's  LocdioM  of  Haihvays. ..     Small  >-vo, 

Wlicclcr's  Civil  EiiitinecriiiL' '*vo, 

AVilson's  Topographical  Surveying Svo, 

Wolfl's  Windiniil  a-  ;i  Prim.'  M.-wv 8vo, 

HVDRAII.ICS. 

[S^eohry  K\c,im;i:i{Im.,  p.  7 
Bazin's  Experiments  upon  the  Cunlraciion  ol'  ihc  Ijijuid  Vein. 

( 'I'l  ant  wine. ; ■^vu, 

Jii)\(\  -  Trciiti-^e  on  I[yiirauli<-s tlvo, 

Cliurch>  .Mccliauics  of  Engineering,  Solids,  and  Fluids. .  .  .8vo, 

Cottiu's  (Jiiiphical  Solution  of  Hydraulic  Problems...   . .  .lamo. 

Fenel's  Treatise  on  tlie  Winds,  Cyclones,  anil  Tornadoes. .  .8vo, 

Folweli's  Water  Supply  Eii^'ineering .Svo, 

Fri/eds  Walerponer nvo, 

Fuertcss  Water  and  F'nblie  Ileal th lim". 

Water   Filtration  Woiks l','ino 

liangiullel  iV   Kultei  s  Flow  ol  Water.     JleiingvV  'I'rantiv  inc. ) 

8vo, 

Hazen  s  Fillnilion  of  Puliiic  Wtiter  Supply >vo, 

HH/.lehursl  s  'lowers  and  Tanks  for  Cities  anil  Towu.> 8vo, 

llerscliel  s  11.")  K\pei  Imeiits     Hvn. 

Kiersled  s  Sewa;:e  Disposal I'imo. 

.Miiviii  <  Wtilei  Supply Svo. 

Extaniuiilion  of  W.-ilei liino, 

Merriiuan's  Tie.'ili«e  oi!  II\  'tnuiliis sivo 


'■> 

IH) 

.*) 

t.() 

4 

00 

."» 

Oi) 

•> 

IMI 

.1 

.■)') 

0 

00 

o 

00 

»') 

in) 

i\ 

'•>) 

'l 

OI) 

') 

.■)0 

'> 

,")0 

1 

e.i 

t 

00 

■") 

on 

5 

00 

4 

00 

3 

50 

3  00 

.^^,00 


(i  lill 


•J 

oO 

4 

00 

4 

(JO 

r, 

00 

1 

>> 

."ill 

■'lO 

4 

00 

l> 

00 

0 

50 

t) 

(10 

1 

-'.5 

.*) 

110 

■t 

00 

NichiiK's  WutiT  r^upply  (CliL'mical  uiiil  t^aniuiryi 8v<>,  fJ  "jt* 

Siliii}'liT'«  IJcsfi  voirs  for  Iiriiratioii Large  8vn,  ,>  (x) 

Tiirrii'iiurc  iiini  Hu-siOl's  Public  WiiUT-supplios. 8vo,  't  00 

Wegmmin's  Waicr  Supply  of  the  City  of  New  York 4t(),  ID  DO 

\V«i;,li;it!r.s  ll\(lruulics.      (l)u  Hois.) Svo,  5  dO 

Whipple's  Mitroscopy  iif  DriiiUiiig  Water fvo,  ;j  50 

Wilwu's  Irritriitiou  Kiiiriiiei'riii;; 8vo,  4  00 

"        Ilydriiulii- luui  I'liicfr  Mining I'Jiiio.  2  00 

Woltl's  Wimlniill  as  ii  I'rinu-  Mover Hvo,  ;!  (X) 

WikkI's  Theory  of  Turbiues !<vo,  2  oO 

LAW, 

Davis's  Klumeulsof  Liiw Svo,  2  50 

Treatise  on  Military  Law ^vn,  7  00 

Sheep,  T  oO 

Manual  for  Courls-iuartial ICmo,  morocco,  1  5u 

Waits  EiiL'iiiceriug  and  Arcliiteelural  .Tuiispniiieuce 8vo,  0  00 

Sliei'i),  6  50 

■      Law  of  Contracts 8vo,  3  00 

Law  of  Operiitioiia  Preliminary  to  Construction  in  En- 

ffiuecrini:  and  Aichitccture svo,  5  00 

Sheep,  5  .jO 

Wimlirop's  Abriilgnient  of  Military  I^mv 12mo,  2  50 


X 


b::| 


MANLFACTLRE5. 

Allen's  Tables  for  I  run  Analysis )»To, 

Heaumont  s  Woollen  uiul  Woisled  Manuladurp 12mi), 

Holland's  Kncyclopa'dia  of  Founding  Terms 12mo. 

'I'lir  Iron  F. Hinder 12mo, 

Supplement, 12mo. 

Eissler's  Kxplosives,  Xitroglyeerine  and  I>ynaniite "^vo. 

Foicl  s  IJoiler  Making  f;,r  toiler  Makers ISmo, 

Metciiife  s  Cost  of  Muniifaelures Svo, 

Metc^aif's  Steel — A  Manual  for  Steel  Users 12uio, 

*  Keisiirs  (Juide  to  Piece  I)yeiiig !Svo, 

Spencer's  Sugar  Manufacturer's  Handbook  . .  .  .16mo,  morocco. 
Ilaiidliook    for    Clicinists    of    Beet    Sugar    Houses. 

ICnio,  morocco, 

Thurston  s  JIunual  ol  Steam  iJoilers Svo, 

Wulke's  Lectures  on  Kxplosives.  ...  Syn. 

'West's  American  Foundry  Practice. .....     .  .    12nio. 

.Moulder  s  Text  book     12ino. 

Wieehmanu's  Sugar  Aiuiiysis Small  ^vo, 

Wuodburv  s  Fire  Protection  of  Mills bvo, 

10 


;i 

Ol) 

1 

.-)0 

3  00 

o 

50 

o 

50 

4 

DO 

1 

DO 

T) 

DO 

.1 

00 

■■ 

00 

•  t 

DO 

n 

00 

5 

DO 

\ 

OD 

o 

r.i> 

■"> 

r,D 

o 

no 

50 

MATERIALS  OF-   ILNCJINEERINU. 

(Ste  iiUo  Km.inki.kinc,  p.  '.) 

Baker's  Masonry  Conslrucii.iii "^"^  *^'  2^ 

Bovv's  Strci.irlli  c.f  Mi.trri;ils ^v..,  .  M 

Burr's  Elaslicily  iind  lUsisluiice  of  Miitfriiils 8vo.  5  00 

Byrne's  Ilidiw.'ivConstriiclioii ^^o,  T)  00 

C'luinh's  jfedmuics  of  Ei.ginecring-Soli.ls  uud  Fluids 8vo,  G  00 

Du  Buis's  Sitnsscs  in  Framcil  Structures Small  4lo,  10  00 

J. ill II son's  Mut.riuls  of  Construction 8vo,  rt  00 

Lanza's  Applir.I  M.Tlnvnics 8vo,  .   oO 

Marlrnss  Tistini,'  Maf'Mials.     iITcnnin!:.) 2  vois..  8vo,  .  .>0 

Mirrill's  Stoiiis  for  BuiMin-  and  IVcoration Svo,  5  00 

Mtrriniau's  Mechani.  s  of  Materials 8™,  4  00 

Sticugth  of  Materials 12ui".  1  ^ 

Paiton's  Treatise  on  Foundations 8vo,  5  00 

Ro.Uwell's  Uoads  atid  Pavements  in  France I'^m".  1  23 

Spaldinir's  Roa.ls  and  Pavements l'-3nio,  2  00 

Thurston's  Materials  of  C'onslnicti<m 8vo,  5  00 

Alulerials  of  Encitieering 3vols..8vo,  8  00 

Vol.  T  ,  Non-melallie   8^0,  '2  00 

Vnl.  IT.,  Iron  and  Steel '*^''-''  ^  :^ 

Vol.  TIL,  Alloys.  Brasses,  and  Bronzes 8vo.  2  50 

Wood's  Resistance  of  Materials *'V0,  2  00 

.MATHC.MATICS. 

Baker's  Elliptic  Functions •  -^vo,  1   50 

*Basss  DiUerential  Calculus l"-'"".  •*  ^^ 

Briggss  Plane  Analytical  Geometry 12mo,  1  00 

Chapinans  Theory  of  Eipiatioiis 12mo,  1  50 

Compton's  Lo^rarithmic  Computations 12mo,  1  oO 

David's  Tntrodnction  to  tlu'  Logic  of  Algebr 8vo,  1  oO 

IlaUted's  F.lements  of  Geometry 8vo,  1  ^a 

"        Pvntlietic  Geometry ^^'"-  ^  '-'" 

Ti          •    r,,.,. .  Tr.u.ii,.'                                     12nio,  100 

Johnson  s  C  urve  1  racmi: 

Differential  Kiiuiitioiis— Ordinary  ni.l  T'attial. 

Small  8vo,  3  50 

..         Tnte-ialCal,'ulus l'2nt",  1   50 

T'nabiidged.     Small  8vo.    {fn  pres*.) 

..         Leas.  Squares l'2mo.  1  .50 

*I  udl<.«  -s  L...^,ri!lunic  ai:d  Other  Tables.     (Bass.) 8vo,  '2  00 

./    ■•         Tri-mnmetry  ^vith 'I'abh-.     (Bass.. 8vo,  3  00 

'Miiini.'s  Ileseiiptive  Geometry  <  Stone  Cultiiig,   8vo,  1   50 

Mei rimau  and  Woodward's  Higher  Mathematics 8vo,  5  CO 

11 


f  i  r. 


1 

^I'l 

1^' 

i:     ^: 

^^^KV 

Hpiif^Hi| 

|Bi.^.V| 

^^^^H^    III  1 

Mcrriiniirs  Mctlin,!  nf  b-;isl  S.iinre=i ^*vo,  |'.'  00 

Rit'e;iii«:  ..'•.;liii.-"ii'-   I)iiTiTfiili:il  ;iiul  Integral  Calculus, 

0  vnls.  in  1,  snmll  Hvn.  '.'  "m 

••                  DiiTcreiitiiil  Ciilculus SinuliNv.i,  3  tM) 

"  Abii.L'iiHnt  (if  IKflL'1'eiitinl  Calculus. 

Sm.ii;  Sv..,  1  .-,0 

Tottt-n's  MelroldL'v ''^^".  2  ."0 

Warrcu's  l-c»ci  iptivr  Cr.iiuctry 2  vc^ls.,  Hvo,  ;!  50 

l>rafiiiii:  lu-iiuiMciits I'Jnio,  1  i") 

Frcc-lKind   Drawing l-.!iii.>,  1   ^0 

"         I. incur  I'rr-iH  .tivc I'Jiiio.  1   ik) 

"         Priniarv  (I.  .inu!;y 12ii:ci.  T"> 

rianc  I'i,.l  ■.cm-     , l-'iii.i.  1,'.") 

"         PnililcMis  an.! 'riicorcins ^<V(^  ','  .".0 

"         Pmji  cti"i;  DrawitiL' ]'2nui,  1    "ii) 

WooirsC'D-iinlinati-  tl.oiuclry Nvo,  2  •"' 

Trigoniiiutlry 12tiui,  1  i>0 

Woolfs  Dc.-^tiijtlivc  <;...ni.liy Large  r<vii.  3  uO 

■MECH.-VMCS-MA*  IIIVHRY 

>,,•  ilhi)  KN(.INKI-.l;lNli,   [i.   7.) 

Balilwin's  Stpnm  ITcaiiiig  fur  liuiMings 12mo,  2  50 

liarr-  Kiucniati<^  vi  Machinery bvo,  2  M 

Bcnjaniiii's  Wrinkles  ainl  llccipfs 12:iio,  2  (M) 

Cliiinlars  Lptlcr.s  to  Mechanic, 12iiio,  2  <» 

Church  s  ^Ifclmnics  of  En.'ineciing 8vo,  6  00 

"         X(il(s  I'.nd  K.vanip'.e^   ii;  Mi-clianics Svo,  2  00 

Crc  hi.ie-  Mechanics  of  tile  (Jinlir 8vo,  3  IH) 

Ci.  inwi  Us  1!< '.Is  nil. 1  Pulleys 12mo.  1   ."iO 

Tdoili.d  Gearing 12mo.  1  .-.O 

Cimiiiioti's  Fir-.t  I.e^xiiis  in  Mct.'il  Working ]2nio,  1   "o 

Cc.in|it..n  aiiil  I)e  I',! li'-i  Speeil  Lathe 12iiio,  1  ."0 

Dana'-    I".lcnieiilar\    Mim  hank's 12ino,  1   .IO 

Oinuey'-^  Machiiii  r_\    1' illern  Makins. I'.'ino,  2  IH) 

*  Dredge's     Tran<.      Kxliibits     P.uihling,     Wnr'ul      Kxposition. 

iMTge  4lo,  half  nioriicid,  ."i  00 

l)u    liois,  Mei  liauii  s.      Vol.  I..   Kineiiiaties 'Sv  i,  I!  "0 

••                    "                    Vol.    II  ,    St,ali<< t<Vn,  100 

>>                '•                Vol    III..  Kinetics 8vo,  ;i  .'.0 

Fil/gerahlMM.^Ion  Machinist IHnio,  1   tH) 

p'lather's  I)\  iciinniin  ler» P2ino,  2  HO 

Itope  Diivjii;; ]2nin,  2  "O 

IlalTs  Car  I.uhrication P-'mo.  1  00 

lioKj*  Saw  iiliug ISmo,  76 

12 


*  .I.linsoii's 'riu'oreticiii  Mcclisiiiits.     An  Eleiiit-ntiiry  Trwitise. 

IJiiio,  fi  1)0 

Jones- Miiciiiiie  I>c-i^ii.     I'liil  I  .  Kiiuiniitiis ^vu.  I  50 

l':iil  11  ,  Sirtiiglli  lUiil  Pruporlioii  i>f 

.\lachi:  (■  I'lirts 8vo,  3  00 

Lanza's  App! ii'ii  M.-(!i,'i!i i.s ^ vu,  T  :.0 

Mncr<i:irs  Kiiicii]:aii> "^vo,  5  00 

Mcnimaii'-  Mccliaiiir.-  uf  .Mnli rials 5*vo,  4  dO 

Mftcalfes  Cosl  of  Maimfactuics «vo,  ■'J  uO 

•iliciiic's  .'.iKilytical  Mfdiauics Hvn,  4  00 

Uiiliaiil-.-  ('onipn--ia  Air rJmc.  1  M 

l!,.l.irisoii's  I'l-iiuiplcs  .if   MiThaiiisiu , f^vo,  ;i  00 

Siiiiiii-  I'rrss.uuikiii.;   ..f  MiliiN t<vo,  ;'.  00 

Tlim   Ions   Fii'  linn  aiiil  1,^-1   W  niU Svo,  ;i  00 

Till'  Aiiiniiil  a- a  .Marhiiit' I'-'mo,  1  OO 

Warnii's  Ma' liiii'-  (oiisiniriiou •Jvn;s.,svo,  ',    "lO 

Wfi-sl'aiir- llulniulics  aii.l  IIyiliaulioM.-ti.r-.     il>u  B.iis.  i.svo,  .">  oO 
Mniiaiiics    of    KiiuiiR'friiii;.       Vol,    III  ,    I'aii    1  . 

Sec.  I.      (Klrin.' ^vn.  ■.  00 

Wii-liai'liV    Mi'ihaiii.  -     of    KiiL'inri'i  in:;.      Vol      111.,    I'aM    1.. 

S.i.ll,     iKiiiii.i '^vo.  5  (JO 

Wfi-liai  h-  Mcarn  Kiiijiiics.     (Dii  IJois.  i «vo.  5  00 

^Voo,!  -  .Vnalyliial  .M.Mliaiiics f^vo,  i!  HO 

Kli'imiitai)  Mtiliaiiics l-iiio,  1  ',5 

"               "                  "           Siippli  iiR-iii  anil  Kt-y 1,'mo.  I  25 

MET  ALLlH'iV. 

AlUii's  Tables  foi  Ii on  Aiiilys: Svo,  n  00 

E.i;l<siii|i's  <'"l'i  ami  Miiiiin Large  ^<vo,  7  .V) 

Ml  lalluru-y  of  >ilvcr Large  Hv.).  7  oO 

•  Kcrl-  Ml  lallurL'V— Sit'cl.  Fnil.  .  tc Hvo,  V)  00 

Kii!ili:iriir-  Ore  l>iis»iii-  ill  Kuropi- -Svo,  1    .'iO 

M.iD.lf  -  .    Ill  -  A  Manual  foi  Sht|  L  hits l-nio,  2  I'O 

O  :»il-ro!l  s  •rivalini  III  of   (liml  OicH f'vo.  '.'  (Ill 

Tloii-li'ii's  lion  aii.l  Sliil    ^vo.  ;!  .".0 

.\llo\-      t<\-<.  :;   Ml 

WiNon'-  (Naniilr  l'io,(-«,-            TJiiio,  IT.tt 

MIMiWAHMiV    AM)    MIMN<i. 

Barriiigcr's  \!liu'rals  if  (  iiiiTciKl  VhIiis  .  ..Oldoiig  iiioinrro,  U  50 

Hiiml'-  Vri.lilalion  of    Mm.s l^mo,  2  .'HI 

I!om1  .  I!i  -ouin  -  of  Niiill,  W.-lMii  VirsriniH >^V(>.  H  IMI 

Map  of  ^,.llll,  \Vi  -I.  1.1  Virijlliia I'ockrl  l.ixik  form,  'i  00 

Hiii-'i  ami  rtiilidil  1  1  >!  Kiiiiuiativc  Miiu-ialogy      N<  «   Kil    '■*<■".  4  00 

13 


m 


^ 


V    N 


Chester's  Catalogue  of  Minerals 8vo,    .»'.  2" 

Paper.  5() 

T)i(  tioiiaiv  iif  llii- Naiiii  s  ..f  MiiKT'il-.     8vo,       3  (X) 

D:.i.  .'s  AiiR'i ic-iii  Loculiiifs  (if  Mincnils. . . .         LiuirpHvu.       1  00 

Dfsci  .plivf  .Miui-r;il(ifry.  (E.  S. )   Large  i^vo.  liaif  liiomcid.      Iv!  W 
Kiist  Apr'L'i'iix  '"  J^.vstcin  of 'liiicralogy.    . . .  Laigc  8v.,,       1  UO 

"      Mill,  ralugy  ami   I'ctrogn-J'y.     ;J.  IX) liiiio,       2  00 

■'      MiiiiiiiN  .■iiiil  IIiiw  til  Siiiiiy  Tlii'iii.      (K.  S) I'Jiiui,       1   5() 

Ti-xt  l.iMik  ,,f  Miiii-Miliigy.     a:.  S.  )...Xt\v  Kditii^ii.     sv,,,       4  IM) 

*  nriiil%n>'r!iiiiiillirii;,  Kxplosivi's,  Compoiiiiils,  ar.il  Roc  k  Diills. 

4tii.  lialf  iiuiiiKiii, 

Eglesliin's  f'ataliiiri'.    nf  Minciui- ami  '^yiii'iiyiii-; Mvo, 

Eis,-lcrs  lAiilosivi'-- -NilroLrlyccritic  and  I>yiiaMiitr Svo, 

HiisMik's  Ruck  fi Milling  Minerals,     it^iuilli. ) Small  8vo, 

Ihlseiig''^  Manual  of  .Mining ."<V(i, 

Kuiilianlrs  Ore  I>re>siiig  in  Kumiie 8vo, 

D'Hii-eiiU's  'I'n  aliiiint  of  (iolil  Ores '^v,., 

*  rVni;,  Ill's  Rcconl  nf  Mineral  Tests I'a|.er.  s- 

R.'^.iiliiiv,  h's     .Mieniseopieal     Pliysiograpliy    of    MiiieraN     ami 

Klirk>.      iMdilgs.  1 ." " Hvi.. 

Saw  \er'>  .\c cidi  nl>  in  Mines Ijirsre  Hvo, 

i5loel^lllidgl■'s  Rcii  Us  and  Snil-i 8vo. 

•Tillman's  Iinpcutaiil  Minerals  and  Rocks 8vo, 

Walke'>  Ler lures  on  K.\ plosives gvo, 

WillianisV  l,illioliMj_v 8vo, 

Wilson  .^  Mine  Venlilalion 12nio, 

"         Ilydraulii  andPlaii-r  Mining 12nio, 

STEAM  AND  ElECIRICAL  ENGINES,  BOILERS,  Etc. 

(.•ve  ■i./.i)  Kn(.IM1  lil.N(.,   p.   7.) 

Baliiwin's  Strnni  IIi  aiing  for  Iluilding- I'Jiiiof  2 

Clerk  s  Ciis  Kiigim Sliinll  Mvo,  4 

Ford  s  lloiler  Making  for  lioiler  Makers ]8!iiii,  1 

1I«  men  way 's  Indiratcr  Piaetii  e !2mo,  2 

Kinl  >.  Sli  Bill  Iioiler  J^leonoiiiy hvo,  4 

Kneiif.-  »  I'rai  Ii<  e  and  'I'lieory  of  llie  Iiijeetor 8vo,  1 

M ill's  Slide  ValVf gvo,  2 

M.  >ers  JIimIi  rn  Loeoinollvc  Cinislnielioii 4|,,.  ]0 

Peulioily  aiiii  Miller's  Sleam-lioilers 8vo,  4 

PcuImkIv's  Tallies  nf  Saliirateil  Steiini 8vo,  1 

"          Till  iiiiod\  miiidis  nf  I  lie  Sieaiii  Engine 8vo,  .'» 

"          Valve  (iiars  for  the  Sli  am  Kngine Hvo,  2 

"          Manual  of  tlie  Sli  am  engine  Indicator I2mii.  1 

Pra\  -  Twriily  ^'eais  »  ilh  llie  Imla  aiiii Large  Hv<i,  2 

14 


r. 

00 

o 

ra 

4 

00 

o 

00 

4 

00 

1 

M 

o 

III) 

.-)0 

5 

00 

* 

00 

I) 

5f> 

o 

Oo 

4 

(H) 

3  00 

1 

25 

'> 

50 

50 
00 
00 
00 
CO 
50 
00 
00 
00 
00 
00 
50 
.10 
50 


P;,piM  una  ()>!■  ibciis's  Tii>'nu(Hlyiiiiiiiit's '--'i"'. 

Ui-aL'iiu's  Sloiim  ami  Klcrtric  l.ncoiiiolivcs I-'""- 

KiMagcn's  Thcriih.ilyiK.ni''  >■     i  Pu  Bois. ) •^^■" 

Siiicluir's  Looiiiiutive  UiniiiiiiK' •• Uiiin. 

Suou.-,  Steiim-iioilfr  Piiicticc •  j 

Thursloii's  Boilei-  Ksplosimis -^""' 

HusiiiK'  nun  I5o:'n--r  Trmls ^^'"' 

Mruur.u  ..f  tlu-   Slmm   Ki.Liiiu.       I'arl    I.,   Sinirtuve 

:iiiil   1  Ijf.iry   

Mamiui   el    lljr    Sl.-Hin    Khlmir'.      rail    11  ,   DcMi-'u. 

(•,.iistrti(ti"ii,  .iiid  oi"  rati. 'II ^v,., 

'^  liiirts, 

"  I'liilnvcipliv  of  !lii'  Sliani  KiiL'ini' I'-'iiio, 

U.-lUclmn'.M,  Ih.-  Mnlivr  i'nu,,    of  ll.at.     iCarnot.l 

i<  ^iiiiiiiia!  v  SicMiii  Kir-'iiius   "^"' 

..  Sicaiii  l.oili  1  (  ..|.MIU(ti..ii  H  h1  t>[.L-iaticill '^vo, 

SpariL'lt'r's  Valve  Ci'ars -^     "• 

N,it(>  on  TlifimailyiiaMiics I'-mo, 

\Vri«liii(ir-  Sli'ani  Kii.i-'inc.     '  U"  Bois) f^^'" 

Wliillmm's  Sloani-cnsrinc  Dcm-ii   •-'■'' 

WilMm'sSlcaiM  I'.oiUT-.     'Kl-iili.T,,         '-'"". 

Wood's  Til.  rin...l}  liami.-    lie;<l    M.Ooi  -,  vtr ^vo. 


*1 

■.'") 

'J 

III! 

•") 

00 

2 

Oil 

00 

1 

5n 

ri 

1)11 

D    IM) 


1(1   (HI 


1  "i" 

■J  .-.(> 

r»  00 

1  00 

5  w 

5  do 

•i  ")0 

4  (II) 


lAlU  r:S,   N\»;i<iHT5     AM)   Ml. ASlVi:5. 

Ailriancc's  Laboratory  taltulatiotis I'^iim, 

Aiitii's  Tabic*  for  Iron  Analysis •^^■" 

Uixby's  Orapliiial  Compiiliug  Tables Mu-il. 

Complon's  Loi;!iritlinis •-'""■ 

Cramiahs  Uaihvav  nn.l  Kaitliwork  Tables t^vo, 

Davis'    KU-vation  ami  Sliuiia  Tables Siimll  f-vo. 

FisUel'-  Tnble  of  Cubir  Yai.N   (ai.lboaia, 

Huil^ou's  Kxeavation  Ttibl.-      Vol.  11 ^^■"' 

Jolmson's  Sludia  anil  KailnworU  Tables '•v... 

I.u.lloM's  Lo-alirnniie  uiul  »  Mlier  Table-.     fHass., l'.Jiiii>. 

Tottcn's  Metrology '^''"' 

M.MIl.ATION. 

nnldwiu'x  Mean.  IC.atini; I-'""' 

IJeanl's  Venlilalioi.  .if  Mims '-'"."• 

Catpenter's  Heiilin^'  ami  VenlilnlInK  of  UulUliugs .Hvo. 

Ocrhartl'x  Sttnllury  House  Inspection !''">"• 

■^IVlUoii'i.  Mine  VenliliUl.m I'"'"' 

n 


1  25 

;■>  no 

1  .-,11 
1  ."ill 

1   oil 
!    oil 

1  •^•l 

'.1    (HI 

2  .V) 


2  .-(•I 

2  .VJ 

1  (III 

1  •:.< 


Ml': 
il: 

r 


MISChLLANEOL'S  PL'BLICAirONS. 

Alport's  Gcni.«.  SniliiiH  III,  LiMiL'ua!:c Gill  fdffes,  $5  0» 

Emmuns  Gi-olotrinil  (Juidu  .i,»,k  tit  Ibc  Kci-ky  Mruiiliiiiis,   mvo.  1  50 

Fern  r>  TR-;ai.-,e  on  the  Wiua-   ""'.  ^  ^'^ 

Hiiint!.s  Aa(l^L•s^:l•s  Drlivereil  b-foie  tli."  Am.  liy.  A>>n.  .  ,1^iik>.  'J  .n) 

Motfs  Till-  Fiillacy  "f  tlic  rrcsciit  TlR'oiy  of  Soiiii.l     S4    ICiiio,  1   ''0 

RHmnls's  Cost  c.f'LivitiL' • -'""■  ^  *'^ 

lii.kells's  llistcry  of   llni—hic  r   r..lylcfliiiir  InsliluK'. .  .  .   ^-vn,  3  00 
Ut.tlKiliiiiii'n    TUu     New     'l\st;iliRMil     t'rilically     KiufliaM/ia. 

IJiii...  1   oO 
••               'llir  Eiiii>liH-i/.ca   N(  w  "I'lNi.      A   i:(W  trsui-Iatioii. 

Liiri.'c  Mo,  2   Or. 

Tutteu's  All  liii;.oiiaiit  (^u-li..ii  in  Mi  !!.>!. ...'v f^vo,  2  00 

Hi;l4Rr:\\    AM)   CMAl.Dr.l;    TllXTIJOOKS. 

]-,,UMi:  >    AM)    'I'm. "I  ■"■KM      >l   M1.N.MJV>. 

Ge^Pllius's   llcbicw   iiiul    fhaalfi'    Ixxiinti    10   OM    ■r.-tainnit. 

iTrcelles.) Small  -llo,  lialt  mcrncro,  5  00 

Green's  Eli'imiitiiry  llibicw  Graiiiniar U'liio,  1  '^5 

Grnmiiiiir  of  tlio  llrliMW  I.aii^'im^'i' (New  Kilitimi  l.Hvo,  3  00 

llobriw  Chiest.inmtliy ^v».  2  00 

Letteris's    Ucbrcw   liiblc   uMusMirctic   -Notts  in   F.iinlislii. 

bvd,  Hiiibfsiiuf,  2  '^5 


^■^ 


,Mni)ICAl.. 

naiiiniirst.irs  lMiy-ii)l().iri('ul  Cbemislry.    iMnndcl.) 8vn.       4  00 

M.itt  s  CimiiMi-iiii'ii,  nii.'1'slibility.  iiiul  Nutritive  V.-iluf  of  Food. 

Liirire  niounlcd  eliart,       1  25 

Hud.ii:iiaii's  Incompatibilities  in  I'le'criptions *?vo.       2  00 

Ste.  IV  -IreiitiMMiii  the  Diseases  i,f  tlie  Dop i^vo.        »  50 

Woodlmir.H  Military  lly,i:iene 1«'""        ^  -^ 

Worcester's  Siimll   llosiiitiils— EslHl)liKlinifiil  and  Maiiiteiiniii  e, 
ineliiding   Atkinson's   Sugjicstions   for    Hospilai    Aubi- 


A-iiiio,       1  25 


1« 


If 


